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60 Years of the Hodgkin-Huxley Model
Trinity College, Cambridge, UK.
July 12 - 13, 2012
In celebration of the 60th anniversary of the publication of the Hodgkin-Huxley
model of the action potential
This congress is being organized to celebrate the 60th anniversary of
the original publication of the Hodgkin and Huxley model of the
generation of the action potential by the squid giant axon. This
publication and the mathematical model it describes is at the core of our
modern understanding of how the action potential is generated, and has
had profound effects on many fields of biological science and in
particular on computational studies of neuronal function. The congress
will be held at Trinity College, Cambridge which is the home academic
institution for the original research.
The authors
Sir Alan Lloyd Hodgkin OM, KBE, PRS Sir Andrew Fielding Huxley OM, FRS
The model
The sponsors
Locations
Lectures
Winstanley
Lecture Theatre
To reach it, turn into
Whewell's Court from
Trinity Street. Turn right
immediately after the first
arch and climb the stone
stairs. At the top of the
stairs, follow the path
round the ziggurat-like
Wolfson building, leaving
it to your left. Then turn
right into Blue Boar
Court. The Winstanley
Lecture Theatre is on the
east side of the Court.
Lunches and
Dinner
Trinity College
Hall
The Hall is in the centre
of the western range of
Great Court. It is reached
by a semi-circular set of
steps up to the Screens
passage
Poster Session
and Pre-dinner
drinks
Old Kitchen &
Nevile’s Court
The Old Kitchen can be
reached either from the
passage off the Screens
marked 'Buttery', or from
a door in the south-east corner of Nevile's Court.
Plaque Unveiling &
Bertil Hille’s Keynote
Lecture
University of Cambridge
Department of
Physiology,
Development and
Neuroscience
Proceed right out of the main gate of
Trinity College along Trinity Street
into Kings Parade. Turn left into
Benet Street past the Corn
Exchange then right into Corn
Exchange Street. At the T- junction
cross Downing Street and the
Entrance to the Downing Street site is almost directly in front of you.
The red triangle pointing
through a blue cross
indicates the Downing
Street entrance PDN is
circled in red (No.15).
The plaque unveiling
ceremony will take place
just outside and the
keynote lecture will be in
the main lecture theatre.
60 Years of the
Hodgkin-Huxley Model ____________________ CONFERENCE PROGRAMME ..................................................................................................
Day One (12/07/2012)
9:00am Welcome and Introduction and Logistics (All Day 1, morning lectures will be held in the Winstanley Lecture Theatre, Trinity
College). 9:20am Historical overview and context
chaired by James Bower
James Bower, University of Texas Health Science Center, San Antonio. Building a computational foundation for neuroscience: lessons learned and where do we go from here
9:40 am Wilfrid Rall retired NIH, Bethesda, Maryland and Gordon M. Shepherd, Yale
University. From Hodgkin & Huxley to the central nervous system: first steps in building biophysically-realistic excitability into central neurons and their dendrites.
10:20am Daniel Gardner, Weill Cornell Medical College, New York, NY.
Sixty Years of Membrane Current in Nerve: The Revolution Evolves
10:40am Tea Break 11:00am Biophysics and Biology
Part 1: The Action Potential Today Energetics chaired by Hans Braun
11:00am Biswa Sengupta, Indian institute of Science, Bangalore, India. and Martin
Stemmler Bernstein Centre for Computational Neuroscience, Munich, Germany.
Energetically optimum action potentials
11:20am Bruce Bean, Harvard Medical School. Sodium channel gating before and during the action potentials of mammalian
neurons
11:40pm David Attwell, UCL, London Energy use in the grey and white matter of the CNS
12:10pm Lunch (In Trinity College Hall)
1:30pm Unveiling of a commemorative plaque (To be held at the Entrance of Physiological Laboratory, University of Cambridge
Please ensure that you are there a minute or two in advance as there will be a prompt start)
Keynote Lecture
(To be held in the Physiological Laboratory Hodgkin Huxley Seminar Room,
University of Cambridge)
2:10pm Bertil Hille, University of Washington.
The context, conception, and impact of Hodgkin and Huxley's action potential model: 1936-1970.
3:10pm Biophysics and Biology (To be held in the Winstanley Lecture Theatre, Trinity College Cambridge)
Part 2: Molecular dynamics of ion channels chaired by Bruce Bean
3:10pm Indira Raman Northwestern University,
Resurgent current of voltage-gated Na Channels
3:30pm Peter Jonas, IST, Austria.
The tale of the polar neuron
3.50pm Lorin Milescu, University of Missouri, Columbia
Playing with ion channel models in real neurons
4:10 pm Ilya A. Fleidervish, Ben-Gurion University, Beer-Sheva, Israel Shedding light on sodium fluxes and action potential initiation in cortical pyramidal neurons
4:30pm William L. Kath, Northwestern University
A sodium channel model with slow recovery from inactivation
4:50pm 5 Minute Previews of Presented Posters
(To be held in the Winstanley Lecture Theatre, Trinity College Cambridge)
Chaired by James Bower
4:55pm Santiago Archila, Emory University, Atlanta, GA
Experimental and computational evidence suggest possible regulation of synaptic conductance via graded post-synaptic chloride homeostasis
5:00pm Ernest Barreto, George Mason University, Fairfax, VA, USA.
The Hodgkin-Huxley Model with Dynamic Ion Concentrations: A Novel Mechanism for Bursting
5:05pm Maria Botcharova, University College London
Phase synchronisation measures from systems with non-critical interactions can show power laws - a signature of criticality
5:10pm Quentin Caudron, University of Warwick, UK
Convergence of Cable Theory Methods
5:15pm Efrat Katz and Michael Gutnick, The Hebrew University of Jerusalem, Israel
Neocortical neurons possess two distinct persistent sodium currents with different voltage dependence and different underlying mechanism of generation.
5:20pm Stefano Luccioli, Istituto Sistemi Complessi – CNR, Sesto Fiorentino, Italy.
Coherence resonances in the Hodgkin-Huxley model in the high-input regime
5:25pm Mark D. McDonnell, Brett A. Schmerl and Daniel E. Padilla, University of South
Australia
Slope-based suprathreshold stochastic resonance due to ion-channel noise in
phasic auditory brainstem neuron models
5:30pm Andreas Neef, MPI Dynamics and Self-Organization, Goettingen, Germany
Somatic sodium channels account for 2nd phase of action potential upstroke in Layer 5 pyramidal cells
5:35pm Mohammad Ali Neishabouri, Imperial College London
Effects of NaV1.8 Clustering in Small Diameter Unmyelinated Axons 5:40pm Uri Nevo, Tel Aviv University, Tel Aviv, Israel
Neuronal activity, beyond electricity and chemistry
5:45pm Netanel Ofer, Bar Ilan University, Israel
The axonal response to current stimuli and morphological modifications - non-linear analysis of the Hodgkin-Huxley model
5:50pm Frederic Roemschied, Bernstein Centre for Computational Neuroscience, Berlin,
Germany
Transduction effects on temperature-invariance of grasshopper auditory receptor neuron spike rates
5:55pm Jan-Hendrik Schleimer* and Martin StemmlerƗ
*BCCN, Berlin, Germany, ƗBCCN, München, Germany
Phase Reduction of Noisy Hodgkin-Huxley Models 6:00pm Joshua Singer, University of Maryland
Repetitive spiking in an axonless retinal neuron
6:05pm David Sterratt, University of Edinburgh
A very fine spatiotemporal grid is required to determine propagation speed accurately in a model myelinated nerve
6:10pm Nobuyuki Takahashi, Hokkaido University of Education, Hakodate, Japan
Synchronization of nerve impulses coupled with electrostatic capacitance in parallel axons model
6:15pm Yulia Timofeeva, University of Warwick, UK
Integrated neural modelling of calcium and electrical signalling
6:20pm Maxim Volgushev, University of Connecticut
Fast computations in cortical ensembles require rapid initiation of action potentials
6:25pm Robert Young, University of Hawaii, Honolulu
The "Lillie Transition": Modeling the Onset of Saltatory Conduction in the Development and Evolution of Myelin
6:30pm Poster Session /Pre Dinner Drinks (Poster session in Trinity College Old Kitchens and Pre Dinner Drinks in the adjoining
Nevile’s Court)
7:30pm Conference Banquet (To be held in Trinity College Hall) After Dinner Speech
Ian Glynn, Professor Emeritus, University of Cambridge “Early Days”
60 Years of the Hodgkin-Huxley Model ____________________
CONFERENCE PROGRAMME ..................................................................................................
Day Two (13/07/2012) 9:00am Biophysics and Biology
(All Day 2 lectures will be held in the Winstanley Lecture Theatre, Trinity College, Cambridge)
Part 3: Evolutionary considerations
chaired by David Attwell
9:00am Bertil Hille, University of Washington. Ion channel evolution: basic mechanisms and results.
9:20am Harold Zakon, University of Texas at Austin
Adaptive evolution of Na channels in pain receptors.
9:40am Modulation of and by ion channels chaired by Michael Häusser
9:40am Michele Migliore, CNR, Palermo.
On the mechanisms underlying the depolarization block in the spiking
dynamics of CA1 pyramidal neurons
10:00am Susanne Schreiber, Humboldt-Universität zu Berlin and Bernstein Center for
Computational Neuroscience Berlin, Germany
Grasshopper neurons keep cool in hot situations: temperature-compensated
spike rates can be achieved in single neurons at no additional energetic cost
The EBSA Lecture
10:20am Andreas Herz, Ludwig-Maximilians-Universität and Bernstein Center
for Computational Neuroscience Munich
Electrical activity in resonant neurons: From sub-threshold
oscillations to phase precision?
10:50am Hans A. Braun, University of Marburg, Germany.
Conductance-Based Computer Models with Hodgkin-Huxley-Type Neurons and
Synapses Adjusted to Experimental and Clinical Tasks.
11:10am Hugh Robinson, University of Cambridge, UK.
Mechanisms and functions of irregular spiking in a class of neocortical
inhibitory interneurons
11:30am Tea Break
11:50pm Developments in Modelling the HH equations
Chaired by James Bower
11.50pm Idan Segev, Hebrew University, Jerusalem.
From a single H&H spike to a family of spike patterns
12:20pm Cengiz Günay, Emory University.
Simulated compensation of experimental artifacts for Hodgkin-Huxley type ion channel parameter fitting
12:40pm Lyle N. Long, The Pennsylvania State University
Efficient and Scalable Neural Network Simulations for Engineering Applications using the HodgkinHuxley Equations
1:00pm Lunch (Trinity College Hall) 2:00pm Ion channels and computation Chaired by Idan Segev
2:00pm Sungho Hong, Okinawa Institute of Science and Technology.
Adaptive Computation of Neurons with Hodgkin-Huxley Mechanisms
2:20pm Yuguo Yu, Yale University and Fudan University, Shanghai.
Does Hodgkin-Huxley theory need to upgrade for mammalian cortical neurons?
2:40pm Fernando R. Fernandez, University of Utah.
Understanding neuronal input-output transformations in the context of in vivo-
like membrane voltage conditions
3:00pm Michael Häusser, UCL, London.
Dendritic Computation
3:30pm Fred Wolf, Max Planck Institute for Dynamics and Self‐Organization, Göttingen,
Germany;
How Details Matter ‐ Computational Capabilities of Neocortical Networks Reflect the Physiology of Action Potential Initiation
4:00pm Tea Break 4:30pm Stochastics, Noise and Chaos
(To be held in the Winstanley Lecture Theatre, Trinity College Cambridge)
Chaired by Peter Rowat
4:30pm Christian Finke, University of Oldenburg, Germany.
Effects of different noise implementations in a Hodgkin-Huxley-type cold receptor model with subthreshold oscillations
4:50pm Kazuyuki Aihara, Institute of Industrial Science, University of Tokyo.
Chaos and Bifurcations in the Hodgkin-Huxley Equations and Squid Giant Axons
5:10pm Michele GIUGLIANO, University of Antwerp.
Accurate and fast simulation of channel noise in conductance-based model
neurons
5:30pm Lech S. Borkowski, Adam Mickiewicz University, Poznan, Poland.
Multimodal transition and stochastic coherence antiresonance in the
periodically stimulated Hodgkin-Huxley model with noise
5:50pm Alessandro Torcini, CNR, Florence.
Coherent response of the Hodgkin-Huxley model in the high-input regime
6:10pm Henry C. Tuckwell, Max Planck Institute, Leipzig, Germany
Inverse stochastic resonance and long-term firing properties in stochastic
Hodgkin-Huxley systems
6:30pm Meeting evaluation and final comments
Abstracts (Listed in alphabetically order of author’s surname)
Kazuyuki Aihara
Institute of Industrial Science, University of Tokyo
CHAOS AND BIFURCATIONS IN THE HODGKIN-HUXLEY EQUATIONS AND SQUID
GIANT AXONS
Chaos, or deterministic chaos, is ubiquitous in nonlinear dynamical systems of the real
world, including biological systems. Nerve membranes have their own nonlinear dynamics
which generate and propagate action potentials. Such nonlinear dynamics of nerve
membranes has been intensively studied experimentally with squid giant axons and
theoretically with the Hodgkin-Huxley equations. In fact, chaos and various routes, or
bifurcations to chaos in nerve membranes were found both with the Hodgkin-Huxley
equations and with squid giant axons by our group (1, 3-7, 9, 13-15). Moreover, a chaotic
neuron model that qualitatively reproduces chaos in nerve membranes observed in squid
giant axons and the Hodgkin-Huxley equations can be described by a simple 1-dimensional
map (8, 9). Chaotic neural networks composed of the chaotic neurons generate various
kinds of spatio-temporal dynamics with abilities of parallel distributed processing (2), and are
implemented by integrated electronic circuits for possible applications (10-12). This talk will
be to confirm the great importance of the deteministic nonlinear dynamics in squid giant
axons, which was formulated by the Hodgkin-Huxley equations, through analysis of neural
chaos and its bifurcations which can contribute even for development of new technology and
engineering(2).
(1) Aihara, K.(1995). Chaos in axons, in The handbook of brain theory and neural networks,
1st edition, edited by M.A. Arbib, The MIT Press, Cambridge, Massachusetts, pp.183-185.
(2) Aihara, K.(2002). Chaos engineering and its application to parallel distributed processing
with chaotic neural networks, Proc. IEEE 90(5), 919-930.
(3) Aihara, K., and Matsumoto, G.(1982). Temporally coherent organization and instabilities
in squid giant axons,J. Theor. Biol. 95(4), 697-720.
(4) Aihara, K., and Matsumoto, G.(1986). Chaotic oscillations and bifurcations in squid giant
axons, in Chaos, edited by A.V. Holden, Manchester University Press, Manchester, and
Princeton University
Press, Princeton, pp.257-269.
(5) Aihara, K., Matsumoto, G., and Ichikawa, M.(1985). An alternating periodic-chaotic
sequence observed in neural oscillators, Phys. Lett. A 111(5), 251-255.
(6) Aihara, K., Matsumoto, G., and Ikegaya, Y.(1984). Periodic and non-periodic responses
of a periodically forced Hodgkin-Huxley oscillator, J. Theor. Biol. 109, 249-269.
(7) Aihara, K., Numajiri, T., Matsumoto, G., and Kotani, M.(1986). Structures of attractors in
periodically forced neural oscillators, Phys. Lett. A 116, 313-317.
(8) Aihara, K., Takabe, T., and Toyoda, M.(1990). Chaotic neural networks, Phys. Lett. A
144(6/7), 333-340.
(9) Aihara, K., and Suzuki, H.(2010). Theory of hybrid dynamical systems and its
applications to biological and medical systems, Phil. Trans. Roy. Soc. A 368(1930), 4893-
4914.
(10) Horio, Y. and Aihara, K. (2003). Neuron-synapse ic chip-set for large-scale chaotic
neural networks, IEEE Transactions on Neural Networks 14(5),1393-1404.
(11)Horio, Y. and Aihara, K. (2008). Analog computation through high-dimensional physical
chaotic neuro-dynamics,Physica D: Nonlinear Phenomena 237(9),1215-1225.
(12) Horio, Y., Ikeguchi, T. and Aihara, K. (2005). A mixed analog/digital chaotic neuro-
computer system for quadratic assignment problems, Neural Networks 18(5-6),505-513.
(13) Matsumoto, G., Aihara, K., Hanyu, Y., Takahashi, N., Yoshizawa, S.,and Nagumo,
J.(1987). Chaos and phase locking in normal squid axons,Phys. Lett. A 123, 162-166.
(14) Matsumoto, G., Aihara, K., Ichikawa, M., and Tasaki, A.(1984). Periodic and
nonperiodic responses of membrane potentials in squid giant axons during sinusoidal
current stimulation, J. Theor. Neurobiol. 3, 1-14.
(15) Mees, A., Aihara, K., Adachi, M., Judd, K., Ikeguchi, T., and Matsumoto, G.(1992).
Deterministic prediction and chaos in squid axon response, Phys. Lett. A 169, 41-45.
Santiago Archila
Emory University, Atlanta, GA
EXPERIMENTAL AND COMPUTATIONAL EVIDENCE SUGGEST POSSIBLE
REGULATION OF SYNAPTIC CONDUCTANCE VIA GRADED POST-SYNAPTIC
CHLORIDE HOMEOSTASIS In some neuronal networks, homeostatic plasticity ensures that network activity levels stay within normal bounds. Here, we investigate a particular synapse in the pyloric network of the crab Cancer borealis stomatogastric ganglion (STG) using both experimental and computational methods, and examine whether this synapse exhibits homeostatic plasticity. The pyloric network of the crustacean STG consists of an intrinsically bursting pacemaker kernel ¿ one anterior burster (AB) neuron electrically coupled to two pyloric dilator (PD) neurons ¿ which inhibits the follower neurons ¿ one lateral pyloric (LP) neuron and several pyloric (PY) neurons ¿ to participate in a triphasic oscillatory pattern. The lone feedback to the pacemaker is inhibitory, primarily chloride (Cl-) mediated, and comes from the LP neuron. We test whether this synapse (LP-to-PD) exhibits homeostatic plasticity by measuring the conductance before, during, and after a strong voltage perturbation; consisting of either a depolarizing or hyperpolarizing voltage-clamp step of 25 mV from baseline (-60 mV) into one of the PD neurons. It serves as a network activity perturbation because many pyloric activity features (such as cycle period, phases, etc.) were affected along with the PD membrane potential. We observe that in several trials, LP-to-PD synaptic conductance changes in response to the perturbation but surprisingly, this change does not oppose the membrane potential perturbation as we hypothesized. We further investigate what possible network activity effects could arise from this change in synaptic conductance by turning to a previously described (Prinz, Bucher, Marder 2004) model pyloric network database. Using >4 million ¿pyloric-like¿ model networks of Hodgkin-Huxley-type neurons, we analyzed the effect of a change in LP-to-PD synaptic conductance on network-level activity features; cycle period, burst duration, duty cycle, delay, and phase were examined for both PD and LP. Following the computational analysis, we returned to the experimental results to find that the synaptic change we observed was not in the right direction for homeostasis of any of these characteristics. Finally, we find that the synaptic response is consistent with homeostasis of two synapse-specific features: synaptic current and reversal potential. Because this current is dominated by Cl- ions, our results point to a possible graded synaptic regulation of internal Cl-. We built a conductance-based model neuron that uses the internal concentration of Cl- as feedback to homeostatically regulate synaptic currents and find that it qualitatively matches our experimental observations.
David Attwell, Renaud Jolivet & Julia Harris,
University College London
ENERGY USE IN THE GREY AND WHITE MATTER OF THE CNS
I will
(i) briefly review the Attwell & Laughlin (2001) energy budget for cerebral cortex and its
recent modifications;
(ii) present an energy budget for the white matter, examining why the white matter uses less
energy than the grey matter, whether myelination really saves energy, and whether
internodal axons need metabolic support from their myelinating glia;
(iii) present a simple account of how information flow through synapses relates to the energy
used by the synapses and show that, while information flow is maximised by reliable
synapses with a release probability of 1, if there is synaptic convergence then maximising
the ratio of information transmitted to energy used predicts an optimal release probability
that is considerably less than 1.
Supported by the European Research Council, EU, MRC, Wellcome Trust and Fondation
Leducq
Ernest Barreto George Mason University, Fairfax, VA, USA.
THE HODGKIN-HUXLEY MODEL WITH DYNAMIC ION CONCENTRATIONS: A
NOVEL MECHANISM FOR BURSTING
We augment the point Hodgkin-Huxley model neuron to include intra- and extra-cellular ion
concentration dynamics and show that this model exhibits periodic bursting [1-2]. The
bursting arises as the fast spiking behavior of the neuron is modulated by the slow oscillatory
behavior in the ion concentration variables, and vice versa. We analyze this system by
separating time scales: bifurcation analysis of the fast (spiking) system reveals a large
repertoire of bursting morphologies, and analysis of the slow ion concentration dynamics
reveals mechanisms of bursting onset and regions of coexistence of bursting and tonic firing.
This analysis also clarifies the behavior of two such interacting neurons and suggests
mechanisms that may underlie pyramidal cell/interneuron interplay [3]. Our work emphasizes
the critical role of ion concentration homeostasis in the proper functioning of neurons, and
points to important fundamental processes that may underlie pathological states such as
epilepsy.
[1] J.R. Cressman, G. Ullah, J. Ziburkus, S.J. Schiff, and E. Barreto, Journal of
Computational Neuroscience 26, 159-170 (2009); erratum, 30, 781 (2011).
[2] E. Barreto and J.R. Cressman, Journal of Biological Physics 37, 361-373 (2011).
[3] J. Ziburkus, J.R. Cressman, E. Barreto, and S.J. Schiff, Journal of Neurophysiology, Vol.
95, pp. 3948-3954 (2006).
Brett C. Carter and Bruce P. Bean
Department of Neurobiology, Harvard Medical School, Boston, USA.
SODIUM CHANNEL GATING BEFORE AND DURING THE ACTION POTENTIALS OF
MAMMALIAN NEURONS
We examined the kinetics of current carried by TTX-sensitive sodium channels during the
action potentials of several types of mammalian neurons. Using acutely isolated neurons
from various regions of mouse brains, action potentials were recorded in current clamp and
then used as the command waveform in voltage clamp experiments in the same cell.
Experiments were performed at 37 ºC using quasi-physiological ionic conditions. Sodium
current evoked by the action potential waveform was isolated by subtracting currents before
and after application of saturating concentrations of TTX. In hippocampal CA1 pyramidal
neurons and cortical pyramidal neurons, sodium current inactivates completely during the
action potential. Inactivation is already near-maximal by the early falling phase of the action
potential, so that very little sodium current flows during the falling phase. In contrast, in two
types of GABAergic neurons - cerebellar Purkinje neurons and cortical interneurons - sodium
channels never inactivate completely during the action potential, and substantial sodium
current flows during the falling phase. In these GABAergic neurons, there is often as much
sodium entry during the falling phase of the action potential as during the rising phase. The
key difference between pyramidal neurons and Purkinje neurons or cortical interneurons is
the duration of the action potential, which is much shorter in the GABAergic neurons studied.
These "fast-spiking" GABAergic neurons have the property of being able to fire steadily at
high rates. The incomplete inactivation of sodium channels during the action potential
constitutes an extra metabolic "cost" in the form of extra sodium entry but also enables rapid
firing by enhancing sodium channel availability immediately after an action potential.
A feature of TTX-sensitive sodium current in most if not all neurons is the presence of
steady-state, non-inactivating sodium current at subthreshold voltages. This "persistent"
sodium current is present at voltages as negative as -80 mV. Even though it is several
orders of magnitude smaller than transient sodium current, persistent sodium current can
amount to several hundred pA at voltages near -60 mV. Persistent sodium current likely
arises from the same channels that underlie transient sodium current and can be accounted
for by gating models in which inactivation is allosterically linked to activation. Functionally,
persistent sodium current at subthreshold voltages amplifies EPSPs and, in some neurons,
drives spontaneous firing.
Lech S. Borkowski
Department of Physics, Adam Mickiewicz University, Poznan, Poland
MULTIMODAL TRANSITION AND STOCHASTIC COHERENCE ANTI-RESONANCE IN
THE PERIODICALLY STIMULATED HODGKIN-HUXLEY MODEL WITH NOISE
I analyze the Hodgkin-Huxley neuron response to a periodic train of short current pulses of
amplitude I and period Ti [1-3]. When stimuli arrive at the resonant frequency the firing rate is
a continuous function of I and scales as (I−Ith)1/2, characteristic of a saddle-node bifurcation
at the threshold Ith. Bistability at the excitation threshold, associated with the subcritical Hopf
bifurcation, is a property of a non-resonant regime. The interspike interval histogram ISIH
undergoes a sharp transition in the Ti − I plane, in the chaotic regime between mode-locked
regions 3:1 and 2:1, where the notation p:q means q output spikes for every p input current
pulses. If the driving period is below the critical value, Ti < Tc, the histogram contains only
odd multiples of Ti. For Ti > Tc even multiples of Ti also appear in the histogram, starting from
the largest values. The ISIH scales logarithmically on both sides of Tc. The coefficient of
variation of ISIH has a cusp singularity at Tc. The firing rate has a minimum slightly above Tc.
I showed recently [2] that the data of Takahashi et al. [4] confirm the existence of this odd-all
multimodal transition. This dynamic singularity is present in different models of resonant
neurons [5] and should not be hard to observe experimentally. One of its interesting
consequences is the ability to desynchronize a neuronal network by driving resonant
neurons with appropriately tuned periodic stimulus. I also study the influence of noise. In
addition to the well-known stochastic resonance there is a stochastic coherence anti-
resonance, defined as a simultaneous occurrence of (i) the maximum of the coefficient of
variation and (ii) the minimum of the firing rate vs. the noise intensity. It occurs over a wide
range of parameter values, including mono-stable regions.
[1] L. S. Borkowski, Response of a Hodgkin-Huxley neuron to a high-frequency input,
Physical Review E 80, 051914 (2009).
[2] L. S. Borkowski, Multimodal transition and stochastic antiresonance in squid giant
axons, Physical Review E 82, 041909 (2010).
[3] L. S. Borkowski, Bistability and resonance in the Hodgkin-Huxley model with noise,
Physical Review E 83, 051901 (2011).
[4] N. Takahashi, Y. Hanyu, T. Musha, R. Kubo, and G. Matsumoto, Global bifurcation
structure in periodically stimulated giant axons of squid, Physica D 43, 318 1990.
[5] L. S. Borkowski, Excitability of neural oscillators, unpublished.
Maria Botcharova University College London
PHASE SYNCHRONISATION MEASURES FROM SYSTEMS WITH NON-CRITICAL
INTERACTIONS CAN SHOW POWER LAWS - A SIGNATURE OF CRITICALITY We investigate whether power laws can originate from pooling distributions of pairwise phase synchrony measures across a collection of independently evolving oscillator pairs. We calculate phase locked intervals (PLIs) between oscillator pairs and the change in number of phase-locked pairs across successive time points (global lability of synchronisation or GLS), for varying intra-pair coupling. We construct a pooled probability distribution from a large collection of such pairs. We explore the role of threshold definitions for phase-locking in producing apparent power law statistics in data. Our results suggest that caution should be applied to the use of power laws as a marker for the presence of critical interactions in biological systems.
James Bower
University of Texas Health Science Center
BUILDING A COMPUTATIONAL FOUNDATION FOR NEUROSCIENCE: LESSON'S
LEARNED AND WHERE DO WE GO FROM HERE
The mathematical model described by Alan Hodgkin and Andrew Huxley 60 years ago not
only clarified the dynamics of the action potential, it also established, as manifest in this
meeting, a quantitative base for computational study of not only the action potential, but also
the function of neurons and the nervous system as a whole. In principle, this is the power of
mathematical models. The question posed by this talk, at the start of our celebration of this
remarkable accomplishment, is how this legacy and the power of mathematical modeling
can be advanced in the future, in a field that is still largely descriptive and mathematics
averse.
Hans A. Braun1, Christian Finke2 & Svetlana Postnova3,4
1Neurodynamics Group, Institute of Physiology, Philipps University, Marburg, Germany 2Complex Systems Group, ICBM, Carl von Ossietzky University, Oldenburg, Germany 3Brain Dynamics Group, School of Physics, University of Sydney, Australia 4Centre for Integrated Research and Understanding of Sleep, University of Sydney, Australia
CONDUCTANCE-BASED COMPUTER MODELS WITH HODGKIN-HUXLEY-TYPE
NEURONS AND SYNAPSES ADJUSTED TO EXPERIMENTAL AND CLINICAL
TASKS.
A computer model, to be acceptable and valuable for experimental or clinical research,
needs to be designed according to the experimental or clinical task. This requirement was
perfectly fulfilled by the work of Hodgkin and Huxley mid of the last century with an
incommensurable ingenious and exceptionally successful combination of experimental and
modeling studies. Based on experimental recordings of action potentials and ion currents,
they have designed a mathematical model that explains the underlying mechanisms by ion
channel activation and inactivation kinetics. Their major assumptions have fully been
confirmed in innumerable experiments and their modeling concept has set standards in
neurophysiology and beyond. Unfortunately, when facing larger scale problems for example,
interactions between neuronal networks in different brain areas, the classical Hodgkin-
Huxley (HH)-type algorithms soon become unhandy and even have been considered as
“computationally prohibitive” (Izhikevich, 2003). Accordingly, a diversity of simplified
approaches has been developed, especially in the biophysics community, like the widely
used Fitzhugh-Nagumo model. However, this type of models was never fully accepted in
experimental and clinical research. The reasons are obvious. Even when specific
phenomena are simulated sufficiently well, the direct relation of the models’ variables and
control parameters to physiological mechanisms are often sacrificed in favor of the
simplification. In the present work we demonstrate that the original HH equations can
significantly be simplified according to specific tasks when the focus is laid on the
experimentally or clinically relevant measures (see Postnova et al. 2011). Additionally, we
describe how this strategy can be used for the implementation of a conductance based, HH-
type model of a chemical synapse that, despite significant simplifications, can account for a
great diversity of clinically most relevant drug effects (Postnova et al. 2010a). Furthermore,
we show that model’s structure allows easy extensions as well as more detailed
implementation of specific functions whenever it is required. Examples will be given ranging
from the examination of experimentally observed single neuron dynamics (Braun et al. 2011,
Finke et al. 2010) to the analysis of neuronal synchronization phenomena (Postnova et al.
2010b) to a conductance based model of sleep-wake cycles (Postnova et al. 2009). We
expect that this simulation concept will facilitate interdisciplinary research and intensify the
exchange between experimentalists and clinicians on the one side and theoreticians and
modelers on the other side. Simulation tools and practical courses based on such an
approach were successfully introduced as part of the regular curriculum at medical schools
of several universities.
References (recent relevant papers):
Braun HA, Schwabedal J, Dewald M, Finke C, Postnova S, Huber MT, Wollweber B, Schneider H, Hirsch MC, Voigt K, Feudel U, Moss F (2011): Noise Induced Precursors of Tonic-to-Bursting Transitions in Hypothalamic Neurons and in a Conductance-Based Model. Chaos 21(4), DOI: 10.1063/1.3671326
Finke C, Freund JA, Rosa E Jr, Braun HA, Feudel U (2010): On the role of subthreshold currents in the Huber-Braun cold receptor model. Chaos 20(4):045107.
Izhikevich EM (2003) Simple model of spiking neurons. IEEE Transactions on Neural Networks 14: 1569-1572
Postnova S, Finke C, Huber MT, Voigt K, Braun HA (2011): Conductance-Based Models of Neurons and Synapses for the Evaluation of Brain Functions, Disorders and Drug Effects. In: Biosimulation in Biomedical Research, Health Care and Drug Development. Eds.: Erik Mosekilde, Olga Sosnovtseva, Amin Rostami-Hodjegan. Springer, Wien - New York, pp 93 - 126
Postnova S, Finke C, Jin W, Schneider H, Braun HA (2010a): A computational study of the interdependencies between neuronal impulse pattern, noise effects and synchronization. J Physiol Paris 104: 176-189, doi:10.1016/j.jphysparis.2009.11.022
Postnova S, Rosa E, Braun HA (2010b): Neurones and Synapses for Systemic Models of Psychiatric Disorders, Pharmacopsychiatry 43 (Suppl. 1): S82-S91, http://dx.doi.org/10.1055/s-0030-1252025
Postnova S, Voigt K, Braun HA (2009): A mathematical model of homeostatic regulation of sleep-wake cycles by hypocretin/orexin. J Biol Rhythms, 24: 523 – 535
Quentin Cauldron
University of Warwick, UK
CONVERGENCE OF CABLE THEORY METHODS A neuron's dendritic tree forms the majority of its surface area and volume. With different
neuronal types demonstrating such varied dendritic morphologies, is it natural to speculate
as to the extent of the computational function of their geometries. The positions of ion
channels, distributed non-uniformly throughout the dendrites, as well as the extent and
distribution of branching, have a strong impact on the way synaptic inputs are integrated and
hence, greatly affect somatic dynamics. Cable theory can be used to model the diffusion of
current along passive or quasi-active dendritic cables, capturing the impact of the tree's
geometry in an impulse response function. Obtaining this response function allows
calculation of the response to stimulus anywhere on the tree, for any stimulus location, on
dendrites equipped with voltage-gated ion channels supporting sub-threshold
oscillatory dynamics, such as hyperpolarisation-activated (h) channels, and for any stimulus
waveform. Active ion channels can be added to the framework to support dendritic spikes,
as has been successfully demonstrated in the Spike-Diffuse-Spike model. Outside of brute-
force numerical simulation, however, methods for solving cable problems on large,
branching structures prove difficult to use. Hence, we consider three time-domain methods
based on a path integral formalism, and one frequency-domain method for constructing
cable-theoretic solutions on trees with passive and resonant dendrites, and assess their
convergence on real neuronal geometries.
Fernando R. Fernandez and John A. White Department of Bioengineering, Brain Institute, University of Utah, 36 Wasatch Drive, Salt Lake City, UT, 84112
UNDERSTANDING NEURONAL INPUT-OUTPUT TRANSFORMATIONS IN THE
CONTEXT OF IN VIVO-LIKE MEMBRANE VOLTAGE CONDITIONS Over the last few years our lab has studied the input-output relationship in a variety of neurons located within the hippocampal formation. Among our primary goals have been (1) to understand how current- and conductance-based inputs affect somatic membrane voltage, and (2) to understand the relationship between membrane potential and spike output. We have been particularly interested in how single cell oscillatory and resonant properties are expressed in spike output and potentially give rise to coherent network activity. To achieve these goals, we have used the dynamic clamp technique to embed neurons in acute brain slices in an artificial synaptic-conductance environment qualitatively similar to that observed in vivo. Primarily, this has meant studying neurons under conditions of high membrane conductance and noise, whereby spikes are generated by random voltage fluctuations resulting from transient changes in conductance. We have also used dynamic clamp to add and subtract voltage-gated conductance using Hodgkin-Huxley formulism to address biophysical mechanism. The talk will focus on these issues and techniques and how they pertain to layer II stellate cells from the medial entorhinal cortex (MEC). Stellate cell intrinsic biophysical properties have been suggested to play a key role in MEC network behavior (e.g. grid cells, theta oscillations). The transformation between subthreshold membrane voltage and spike output, however, is poorly understood. Moreover, the ability for spike activity to code time varying stimuli (e.g. theta or gamma activity) under a regime were spikes are driven by random voltage fluctuations has not been studied. We have used a variety of measures to quantify the input-output transformation of stellate cells and asked how these properties change as a function of synaptic input statistics consistent with in vivo conditions. Specifically, we have addressed the following questions: 1) How are subthreshold membrane properties affected by changes in conductance associated with synaptic activity? 2) How are subthreshold resonance and/or pacemaking properties transmitted into the spike output of the cell under a regime dominated by stochastic voltage fluctuations? 3) What is the capacity for stellate cell spike activity to code time varying stimuli of different frequencies? 4) How do balanced and correlated synaptic conductance activity influence membrane voltage fluctuations and spike output? In general, we find that subthreshold membrane voltage properties are a poor predictor of spike output activity. Likewise, subthreshold voltage and spiking properties are differentially affected by synaptic conductance activity. These differences are primarily a result of spike refractory and afterhyperpolarization dynamics governed by voltage-gated potassium currents. Finally, the input-output transformation of stellate cells is highly dependent on background synaptic conditions. In particular, correlations between excitatory and inhibitory synaptic conductance shape membrane voltage fluctuations and modulate input-output functions in neurons. These results indicate that a potential contribution of cells to network activity needs to be based on analysis of spiking dynamics and that cells may dynamically adjust their input-output functions.
Related Publications 1. Fernandez FR and White JA (2008) Artificial synaptic conductances reduce subthreshold oscillations and periodic firing in stellate cells of the entorhinal cortex. Journal of Neuroscience 28: 3790‐3803.
2. Fernandez FR and White JA (2009) Reduction of spike afterdepolarization by increased leak conductance alters interspike interval variability. Journal of Neuroscience 29: 973‐986.
3. Fernandez FR and White JA (2010) Gain control in CA1 pyramidal cells using changes in somatic conductance. Journal of Neuroscience 30: 230‐241.
4. Economo MN, Fernandez FR, and White JA (2010) Dynamic clamp: Alteration of response properties and creation of virtual realities in neurophysiology. Journal of Neuroscience 30: 2407-2413. 5. Fernandez FR*, Broicher T*, Truong A, and White JA (2011) Membrane voltage fluctuations reduce spike frequency adaptation and preserve output gain in CA1 pyramidal neurons in a high conductance state. Journal of Neuroscience 31: 3880‐3893. (*: these
authors contributed equally to the work) 6. Kispersky TJ, Fernandez FR, Economo MN, and White JA (2012) Spike resonance properties in hippocampal O-LM cells are dependent on refractory dynamics. Journal of Neuroscience. In press.
Christian Finke, Hans A. Braun*, Ulrike Feudel Theoretical Physics / Complex Systems Research Group, ICBM, Carl von Ossietzky University Oldenburg, Germany * Neurodynamics Group, Institute of Physiology, University of Marburg, Germany
EFFECTS OF DIFFERENT NOISE IMPLEMENTATIONS IN A HODGKIN-HUXLEY-TYPE
COLD RECEPTOR MODEL WITH SUBTHRESHOLD OSCILLATIONS While simplistic (or reductionistic) approaches have their own merits in neuronal modeling (such as ease of analysis and efficiency of computation), models based on the Hodgkin-Huxley-formalism remain the physiologically more reliable alternative and are more suited for a connection of theoretical and experimental studies. In this approach, the state variables have a direct physiological meaning and can, at least in principle, be observed in experiments. Since biological systems are inherently noisy, the challenge of incorporating stochasticity into a model arises naturally. In the conductance-based approach, one immediately has the choice between several variables which yield an interpretation of the stochastic input either as current fluctuations or randomness in the gating mechanisms. The talk will focus on the dynamical effects of these two major noise sources in a mammalian cold receptor model with subthreshold oscillations, namely the Huber-Braun model. The model is, on the one hand, a simplification of the Hodgkin-Huxley model due to some assumptions which reduce the complexity of the voltage dependences. On the other hand, it extends the original model by the introduction of slow, subthreshold currents generating a persistent subthreshold rhythmicity. This leads to a wealth of different firing states and transitions which can be found in many hippocampal neurons as well. The first approach to be discussed in the talk treats noise as an additional current, the second approach assumes the stochasticity to be rooted in the internal conductance dynamics of the neuron. We compare the effects of both noise implementations and discuss the physiological rationale behind them. Furthermore, we explain major differences between the two implementations in terms of state space dynamics.
Ilya A. Fleidervish
Department of Physiology, Faculty of Health Sciences and Zlotowski Center for
Neuroscience, Ben-Gurion University, Beer-Sheva, 84015, Israel
SHEDDING LIGHT ON SODIUM FLUXES AND ACTION POTENTIAL INITIATION IN
CORTICAL PYRAMIDAL NEURONS
Sixty years of intense experimental and theoretical research since the seminal Hodgkin and
Huxley discovery of general mechanisms of neuronal excitability have proven that it is very
difficult to arrive to a consensus model which can satisfactorily account for all the features of
the action potential (AP) generation in central neurons. Ultimately, the controversies reflect
a paucity of available information on Na+ channels at the axonal site of AP initiation,
including their density and dynamic properties. These spatially inhomogeneous
characteristics are hard to investigate only using electrical recordings. We therefore used a
combination of patch clamp and high-speed fluorescence imaging of Na+-sensitive indicator,
SBFI, to measure Na+ fluxes in cell bodies and processes of mouse layer 5 pyramidal
neurons during the sub- and superthreshold depolarizations. We previously showed that Na+
flux elicited by bursts of APs in the axon initial segment (AIS) is about threefold larger than in
the soma. Our present, more precise measurements of the peak of Na+ flux during single
APs revealed that it is maximal (2.3 ± 0.3 times greater than somatic) in a region of ~1/3 the
AIS length which is separated from the soma and from the first myelinated internode by
regions which contain fewer Na+ channels. Compartmental modeling predicts that each of
these AIS regions plays distinctive roles in AP generation: The most proximal one serves as
a resistive “spacer” to isolate the developing AP from the somatodendritic capacitive load,
the middle region is a main source of depolarizing current, and the most distal region, at
which the AP actually initiates, provides sufficient capacitance for AP propagation. Several
studies have attributed the lower threshold for AP initiation in the axon to lower activation
voltage of the axonal Na+ channels. The leftward shift in Na+ channel activation
characteristics is expected, however, to diminish the AIS channels availability in the
subthreshold range of voltages, if their inactivation indeed derives its apparent voltage-
dependence entirely from the activation. Using the amplitude of spike-evoked Na+ flux as an
estimate of Na+ channel availability before an AP was fired, we found that almost all axonal
channels are available at rest. Thus, stepping the membrane even to very negative
potentials caused no significant increase in the AIS Na+ influx. By applying slow
depolarizing current ramps to the soma, we found that the availability of the AIS Na+
channels changes very little (<20 %) by depolarization of up to –55 mV. Thus, the voltage
dependence of inactivation of the AIS Na+ channels is either similar or rightward shifted as
compared with somatic channels, making the formers fully available even at subthreshold
voltages. This conclusion was further elaborated in Na+ flux measurements during the slow
voltage ramps. Flux-voltage relationship associated with activation of slowly inactivating,
persistent Na+ conductance (GNaP) in the AIS was shifted by about 15 mV in the
hyperpolarizing direction as compared to the cell body. The shape of this relationship
indicates that the axonal GNaP is primarily generated by “window mechanism” predicted by
the Hodgkin-Huxley formalism.
Supported by grant 1593/10 from the Israel Science Foundation
Daniel Gardner
Lab of Neuroinformatics, Department of Physiology and Biophysics, Weill Cornell Medical
College, New York, NY USA.
SIXTY YEARS OF MEMBRANE CURRENT IN NERVE: THE REVOLUTION EVOLVES
t – 60
I begin not by looking back 60 years to the 1952 papers, but looking back the same span
from 1952, to find Otto Loewi and Henry Dale, of course, and Erlanger and Gasser, but also
Sherrington, Freud, Brodmann, Alzheimer, Cajal, Bernstein and Hermann and Overton,
even some of Golgi, William James, Hughlings Jackson, and David Ferrier. All of these were
less distant to Hodgkin, Huxley—and Katz—than the 1952 papers are to us today. The fact
that the 1952 papers, especially ‘1952d’, are read and cited and relevant today justifies our
appreciation and our projection of their continued meaning.
t + 20
Only 20 years after, Cole extensively reviewed work leading up to the model and beyond,
observing that by 1972 computer solutions of the equations were no longer novel.
t + 40
By 1992, new methods had informed the model, extended its scope, and identified the
channels. These enabling techniques included among others the identification, purification,
molecular cloning, and site-specific mutagenesis of channels, gating currents, the molecular
electrophysiology enabled by the patch clamp, pharmacology, and more powerful
computational platforms for modeling, as well as the expanded and more general reaction
rate models and parallel work in ligand–gated conductances. This progress led me to
commission invited reviews forming a special Physiological Reviews volume entitled Forty
Years of Membrane Current in Nerve. Rall and Segev and coauthors provided one; others
came from Armstrong, Catterall, Pallotta and Wagoner, Pongs, and Andersen and Koeppe.
Several of us used a revolutionary motif, noting that the 1952 papers had started a revolution
in science, and we noted that the revolution succeeded, continued, acquired adherents,
coexisted with a revolution in molecular biology, and generated anticipation, not just
retrospection. I also said the influence diverges to “form a tree that branches throughout
modern biology.”
t + 60
Today, we note the evolution of this revolution. Successful revolutions create opportunities
for the small, previously-ignored, and insignificant, and these are the many prokaryotes with
crystallizable Na and K channels. Counter-revolutions stimulate reactions from the old
regime, such as the blurring of channels and transporters. I examine significant branches of
the tree:
• conductances, gating, and selectivity and their molecular and structural correlates,
• a multiplicity of conductances, channels, and their functional consequences,
• transporting channels and channeling transporters,
• translational and clinical correlates—the channelopathies,
• new techniques to control membrane current in nerve, and
• model evolution, complexity, and computability and the rise of computational neuroscience.
t –> ∞
Projection suggests future progress in structures and structure-function correlates informing
family resemblances and convergent evolution of channels and transporters, informative
dimensional reduction and computation for both single-cell models and networks, and an
expanding appreciation of action potentials, their modulation, and the informational and
functional consequences.
Hodgkin and Huxley knew that their model and its testability, applicability, and scope would
evolve, and said so. I noted in 1992 that their example “helped convince zoologists,
mathematicians, physicists, physical chemists, engineers, and even computer scientists to
move into physiology and to adopt its methods and goal of functional analysis” and this, like
the model itself, continues.
BIBLIOGRAPHY
Armstrong, C.F. (1992). Voltage-dependent ion channels and their gating. Physiol. Rev. 72:
S5-S13.
Catterall, W.A. (1992). Cellular and molecular biology of voltage-gated sodium channels.
Physiol. Rev. 72: S15-S48.
Cole, K.S. (1972). Membranes, Ions, and Impulses: A chapter of classical biophysics.
Berkeley, CA: University of California.
Gardner, D. (1992). A time integral of membrane currents. Physiol. Rev. 72: S1-S3.
Hille, B. (2001). Ionic Channels of Excitable Membranes, 3/e. Sunderland, MA: Sinauer.
Hodgkin, A. L. (1976). Chance and design in electrophysiology: an informal account of
certain experiments on nerve carried out between 1934 and 1952. J. Physiol. Lond. 263: 1-
21.
Hodgkin, A. L. and Huxley, A. F. (1952d). A quantitative description of membrane current
and its application to conduction and excitation in nerve. J. Physiol. Lond. 117: 500-544.
Huxley A.F. (2004). Andrew F. Huxley. In: The History of Neuroscience in Autobiography, vol
4, edited by L. R. Squire. Oxford:Academic Press-Elsevier, pp 282-319.
Pallotta, B.S. and Wagoner, P.K. (1992). Voltage-dependent potassium channels since
Hodgkin and Huxley. Physiol. Rev. 72: S49-S67.
Pongs, O. (1992). Molecular biology of voltage-dependent potassium channels. Physiol.
Rev. 72: S69-S88.
Andersen, O.S. and Koeppe, R.E. (1992). Molecular determinants of channel function.
Physiol. Rev. 72: S89-S158.
Rall, W., Burke, R.E., Holmes, W. R., Jack, J.J., Redman, S.J., and Segev, I. (1992).
Matching dendritic neuron models to experimental data. Physiol. Rev. 72: S159-S186.
Michele Giugliano
University of Antwerp
ACCURATE AND FAST SIMULATION OF CHANNEL NOISE IN CONDUCTANCE-
BASED MODEL NEURONS
A limit of the Hodgkin-Huxley (HH) work is that it focused on the average effect of the
underling microscopic components of excitability, ignoring the impact of their random
fluctuations. Today we know that random gating of membrane conductances is a major
source of intrinsic neuronal variability, and this can be captured by Montecarlo simulations of
Markov models, accounting for individual channels.
However, generalizations of the HH equations have been proposed to incorporate channel
fluctuations, aiming at capturing effectively the impact of random opening and closing of
individual channels, while retaining a compact formalism and fast simulation times. These
generalized HH equations follow a Langevin-like approach, which involves stochastic
differential equations and no Montecarlo methods.
It took however more than a decade to realize that adding extra noise terms in the HH
equations requires extra care, as the agreement between microscopic and macroscopic
descriptions might be compromised.
In this talk, I will review existing Langevin generalizations of the HH equations and provide a
simple approximation that reproduces accurately the statistical properties of the exact
microscopic simulations, under a variety of conditions, from spontaneous to evoked
response features. I will emphasize that this approximation can be employed for a variety of
voltage- and ligand-gated ion currents.
In conclusion, by the analysis of the properties emerging in exact Markov schemes by
standard probability calculus, I will show the sources of inaccuracy of the previous
proposals.
The recent paper related to this work has been published here:
http://www.ploscompbiol.org/article/info%3Adoi%2F10.1371%2Fjournal.pcbi.1001102
Cengiz Günay
Emory University
SIMULATED COMPENSATION OF EXPERIMENTAL ARTIFACTS FOR HODGKIN-
HUXLEY TYPE ION CHANNEL PARAMETER FITTING
The Hodgkin-Huxley type formalism has been the established form for characterizing
macroscopic channel currents. Although it is now known that they provide an approximation
to the real channel gate movements, they contain far fewer parameters than more realistic
Markov state models. Experimentalists often use exponential fits to Hodgkin-Huxley
equations to characterize channel activation and inactivation properties independently.
However, even in the Hodgkin-Huxley channel model, the activation and inactivation gates
interact non-linearly in composing the final current and it is better to fit the parameters of
these two gates together to obtain more realistic channel parameters. Parameter fitting in
this case can be achieved by replacing exponential functions with simulated computational
models. Currently available computational resources allow performing these fits by iterative
simulations of the model while varying parameters with optimization algorithms (e.g.,
Simplex). Regardless, fitting channel parameters to experimental voltage-clamp data
assumes there are no other artifacts. This is most often not the case because of space
clamp issues arising from channels being located far away (e.g., on a dendrite) from the
recording electrode (e.g., at the soma), the ability to only partially block or inability to block
other existing channels, and the interference from electrode and membrane passive
properties. Passive properties, like electrode series resistance and membrane capacitance
and leak, can be compensated for by recording hardware. However, hardware compensation
cannot be used in certain cases, such as when compensating for electrode series resistance
and cell capacitance artifacts in small cells (e.g., in Drosophila) that cause ringing and
instability of amplifier compensation circuits. Another case is when there is drift in the
electrode series resistance over long recordings, which is not tracked by hardware
compensation because it is only measured when manually triggered. Passive artifacts in
these special cases and the other previously listed artifacts can in principle be compensated
for by simulating models of the artifacts and then mathematically removing them to estimate
the original signal. In the simplest case, for instance in the leak-subtraction paradigm, the
leak current is estimated by a model and then subtracted from the data. In more advanced
cases, Hodgkin-Huxley models of components can be simulated to remove artifact and help
estimate the current passed by a single type of ion channel. We propose using this method
as a straightforward offline computational modeling paradigm for fitting channel parameters
under these non-ideal conditions. We address leak, capacitance and series resistance
compensation in small cells, the effect of electrode artifacts, and also space clamp artifacts
caused by cell morphology. We demonstrate a Matlab implementation of this parameter
fitting paradigm of Hodgkin-Huxley parameters and also suggest other alternatives like using
the Neuron software. Overall, these methods have the potential to increase the efficiency of
finding Hodgkin-Huxley type channel parameters from experimental data and also offer a
solution to previously inaccessible cases like in small Drosophila neurons and drifting
electrode artifacts not tracked by hardware compensation.
Relevant Conference Abstracts
Günay C, Prinz AA (2011). An offline correction method for uncompensated series
resistance and
capacitance artifacts from whole-cell patch clamp recordings of small cells. BMC
Neuroscience 2011, 12(Suppl1):P259
Günay C, Dharmar L, Sieling F, et al. (2011). A novel model of an identified drosophila crawl
motoneuron for investigating functional effects of ion channel type across larval
developmental stages. BMC Neuroscience 2011, 12(Suppl1):P2581
Relevant Papers
Günay C, Prinz AA (2010). Model calcium sensors for network homeostasis: Sensor and
readout parameter analysis from a database of model neuronal networks. J Neurosci,
30:1686–1698. NIHMS176368,PMC2851246
Günay C, Edgerton JR, Li S, et al. (2009). Database analysis of simulated and recorded
electrophysiological datasets with PANDORA’s Toolbox. Neuroinformatics, 7(2):93–111
Günay C, Edgerton JR, Jaeger D (2008). Channel density distributions explain spiking
variability in
the globus pallidus: A combined physiology and computer simulation database approach. J
Neurosci, 28(30):7476–912
Michael Häusser
University College London
DENDRITIC COMPUTATION
The computational power of dendrites has long been predicted using modelling approaches,
but actual experimental examples of how dendrites solve computational problems are rare.
I will discuss results from experiments combining patch-clamp recordings with two photon
imaging and glutamate uncaging demonstrating that the active dendrites of cortical
pyramidal neurons allow them to discriminate spatiotemporal sequences of synaptic inputs
along single dendrites. This provides a dendritic mechanism for pyramidal neurons to
compute direction and velocity, and shows how dendrites can be used to decode
spatiotemporal patterns of input.
Andreas V.M. Herz
Ludwig-Maximilians-Universität and
Bernstein Center for Computational Neuroscience Munich
ELECTRICAL ACTIVITY IN RESONANT NEURONS: FROM SUBTHRESHOLD
OSCILLATIONS TO PHASE PRECISION?
Stellate cells in rat entorhinal cortex exhibit pronounced subthreshold membrane-potential
resonances as well as clustered spike patterns under in-vitro conditions. Both phenomena
can be described at a quantitative level by a resonate-and-fire model that takes the salient
biophysical properties of entorhinal stellate cells into account. In the behaving rat, the same
cells become active in multiple regions of the external world that form a hexagonal lattice. As
the animal traverses one such firing field, action potentials tend to occur at successively
earlier theta phases of the local field potential. This phenomenon is called phase precession.
In this talk, I will discuss the relation between the in-vitro and in-vivo findings, show that
phase precession at the single-trial level is significantly stronger than trial-averaged data
suggest and argue that due to the intrinsic biophysical properties of stellate cells their
multiple firing fields can operate as independent elements for encoding physical space.
Bertil Hille
University of Washington
THE CONTEXT, CONCEPTION, AND IMPACT OF HODGKIN AND HUXLEY'S ACTION
POTENTIAL MODEL: 1936-1970
Textbooks of the 1930s give no definite information on the magnitude or the origins of the
resting potential and the action potential (AP). A burst of discovery in the period 1936-1939,
revealed propagation by local circuit currents, dramatic loss of membrane impedance during
the spike, the magnitude of the resting potential, and the unexpected overshoot of the action
potential. The final breakthrough and synthesis was developed in 1948-1952. The ionic
hypothesis is tested and proven. Hodgkin and Katz (1949) showed that the overshoot
follows the Nernst equation for Na ions. HH measured membrane currents with voltage
clamp. Their special insight was to attribute them to movements of specific ions and to
separate them into Na and K components. As a proof of principle and completeness they
made their model. The mechanism of regeneration and propagation were fully explained.
With respect to mechanisms of the ionic conductances themselves, they made no
commitments. Only in the period 1964-1975 was the concept of ion channels crystallized.
The ionic conductances were due to selective gated aqueous pores of low abundance and
high throughput. The pores were protein. The selectivity filter had a maximum diameter and
used ion radius and charge density to discriminate ions. The voltage gating was driven by
movements of charged voltage sensors. The symmetrical formal kinetics of HH were
recognized as only an approximation of asymmetrical multi-state conformational changes of
these proteins. Only in the 1980s and after did the full versatility, diversity, and evolutionary
relationships begin to emerge through molecular biology.
Ion channel evolution: basic mechanisms and results
Darwin's evolution was based on selection of the fittest in populations of organisms with
natural variation. We know now of enormous population variation at the level of organisms,
chromosomes, clusters of genes, genes, exons, and base-pairs. Ion channels evolved at all
these levels. Although some of the ion channels in our neocortex may seem highly
specialized, the origins of most of them can be traced to brainless prokaryotes. Relatives of
these channels appear in all of cellular life. From the beginning, which is not well worked
out, only a handful of major channel families evolved by gene and chromosomal duplication
together with recombination. By far the largest superfamily includes the tetrameric near-and-
distant relatives of the voltage-gated cation channels revealed by Hodgkin and Huxley. The
next largest may be the pentameric, ligand gated channels used by animals in fast chemical
synapses, and then perhaps the dimeric anion channels/transporters. Almost all we know
today about the evolution of ion channels we owe to genomics.
Sungho Hong Computational Neuroscience Unit, Okinawa Institute of Science and Technology, 7542 Onna, Onna-son, Okinawa 904-0411, Japan
ADAPTIVE COMPUTATION OF NEURONS WITH HODGKIN-HUXLEY MECHANISMS
A primary function of neural systems is processing information in given inputs. A popular
approach to study this is to model those systems as feature detectors that generate outputs
in response to particular preferred patterns in their inputs. The functional model with the
preferred features (or “receptive fields”) can be discovered by statistical analysis of the
inputs/outputs of the system, and this statistical modeling has been successful in
characterizing the function of single neurons and neural networks particularly in sensory
systems.
I will present recent progress in applying this methodology on biophysically defined model
neurons such as the Hodgkin-Huxley model. In computer simulations, we can observe all the
variables we need, and this becomes an ideal ground to study the biophysical origin of the
statistical model and identify which biophysical mechanisms are the substrates of its
components.
The functional model of the Hodgkin-Huxley neuron has a rich structure comprising multiple
features [1]. Furthermore, how much each feature contributes to spiking is determined by a
nonlinear boundary separating the spiking and non-spiking region in the space of stimuli.
Further study using a reduced version of the Hodgkin-Huxley revealed their biophysical
underpinnings: the features correspond to the linearized dynamics of mechanisms such as
Na+ and K+ channel activation, and the boundary corresponds to the dynamical threshold
where the inward current starts to overcome the outward current [2].
The multiple features and nonlinear threshold are important because they can generate
adaptive behaviors even without plasticity [3]. Since the stimulus statistics control which part
of the threshold is mostly used, the most important feature for a spiking response depends
on the context — stimulus statistics. Indeed, the Hodgkin-Huxley and reduced model
displayed such adaptive behaviors [1, 2]. Furthermore, this “intrinsic adaptation” is crucial for
understanding other models based on the Hodgkin-Huxley mechanisms. Neurons with
reduced Na+ conductance and/or increased leak/K+ conductance can have class 3
excitability [4], and show strikingly different response patterns to noise in the input than class
1/2 neurons [5]. We showed that this phenomenon can be well explained via intrinsic
adaptation [6].
Finally, I will briefly present our recent work about implications of this adaptive neuronal
computation in population coding, focusing on noise correlations due to common input.
Previous work suggested that the correlation is simply a function of firing rates, excluding
any possibility of correlation-based coding that is independent of rate codes [7]. However,
we demonstrated that this argument only applies to “integrator” neurons with a single
feature, which can use only rate-comodulation. Conversely, “coincidence detector” neurons
with class 3 excitability can generate nearly rate-independent correlation since their multiple
features contribute to the correlation not only by rate-comodulation but also by sharp
synchronization [8]. This implies that the rich structures in single neuron computation as
seen in the Hodgkin-Huxley model enable diverse coding strategies at the population level.
References
1. Agüera y Arcas B, Fairhall AL, Bialek W: Computation in a single neuron: Hodgkin and Huxley revisited. Neural Comput 2003, 15:1715–1749.
2. Hong S, Agüera y Arcas B, Fairhall AL: Single neuron computation: from dynamical system to feature detector. Neural Comput 2007, 19:3133–3172.
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Peter Jonas and Hua Hu
IST Austria (Institute of Science and Technology Austria), Am Campus 1, A-3400
Klosterneuburg, Austria
THE TALE OF THE POLAR NEURON
The classical view of neuronal function holds that electrical signals are integrated in the
dendrites, converted into sequences of action potentials (APs) in the axon initial segment,
and propagated reliably along the axon to the presynaptic terminals to initiate transmitter
release. However, this simple concept of functional polarity in neurons seems to be violated
in multiple ways. First, APs actively backpropagate into the dendrites. Second, several types
of neurons generate dendritic spikes. Third, EPSPs invade the axon. Finally, AP propagation
in the axon is highly modulated, giving rise to information processing capabilities. However,
our knowledge of single neuron computation is based on the extensive analysis of a
restricted number of glutamatergic principal neuron types, like layer 5 neocortical pyramidal
cells and CA1 pyramidal neurons. Little is known about signaling in GABAergic neurons,
which mediate fast feedforward and feedback inhibition and substantially contribute to
information processing in the brain.
To examine the rules of signal propagation in these cells, we performed confocally targeted
subcellular patch-clamp recordings from interneuron dendrites and axons. Simultaneous
dendritic and somatic recordings revealed that backpropagated APs in the dendrites had
small amplitudes and that dendritic spikes were absent in fast-spiking neurons. Dendritic
recordings further indicated that voltage-gated Na+ channels in the dendrites were present
only at low density. Simultaneous axonal and somatic recordings revealed that action
potentials were initiated at a constant site, ~20 µm from the soma and were propagated into
distal axons with high speed and reliability. Axonal recordings further indicated that voltage-
gated Na+ channels were present at uniquely high density throughout the entire axon (gNa-
bar ~ 200 mS cm-2). Thus, fast-spiking GABAergic interneurons show unique polarity
properties, with Na+ channel density being low at the dendrites and the soma, but high in the
entire axon. Quantitative modeling using Hodgkin-Huxley-type Na+ and K+ channels
indicated that the highly polar channel arrangement along the dendritic-somatic-axonal axis
provided optimal conditions for both high-frequency action potential generation and reliable
propagation of spike trains over the axon. In conclusion, fast-spiking parvalbumin-expressing
interneurons, unlike many types of pyramidal neurons, obey the rules of a “canonical polar
neuron”, with clearly defined dendritic input and axonal output structures.
Hu H, Martina M, Jonas P (2010) Dendritic mechanisms underlying rapid synaptic activation
of fast-spiking hippocampal interneurons. Science 327: 52-58.
William L. Kath
Engineering Sciences and Applied Mathematics, McCormick School of Engineering
Northwestern University, Evanston IL 60208-3125 USA
A SODIUM CHANNEL MODEL WITH SLOW RECOVERY FROM INACTIVATION
Realistic computational simulations of neuronal functionality require incorporation of ion
channel models that faithfully reproduce experimental findings. Using a topology-mutating
genetic algorithm that searches for the best-fit state diagram and transition-rate parameters,
a state-dependent model of the sodium channel present in the dendrites of CA1 pyramidal
neurons was developed. This sodium channel model exhibits both fast and slow inactivation.
Fast inactivation refers to a non-conducting channel state that follows quickly after
depolarization and activation (within milliseconds) and from which channels recover quickly
when the voltage is restored to resting levels [1]. In response to either sustained
depolarization [2, 3] or a train of depolarizing pulses [4, 5], however, the fraction of sodium
channels available for activation can also decrease rapidly, but in this case recovery occurs
much more slowly, in times on the order of seconds rather than milliseconds. This form of
inactivation has therefore been called “prolonged” or “slow” [4, 5]. The presence of such
widely disparate timescales makes creating state-dependent models of these channels a
challenge. The genetic algorithm used to find this state-dependent model searches over the
space of model topologies and the space of rate parameters simultaneously. Furthermore,
an automated, computationally efficient method is used to satisfy the principle of microscopic
reversibility, an equilibrium condition which imposes constraints on state topologies with
loops [6, 7]. The algorithm uses a sequential approach, also known as goal programming [8],
to optimize the model’s output when compared to multiple voltage clamp protocols. Goal
programming allows multiple objectives to be employed without the need to assign separate
weights to each of them.
The optimized model to which the algorithm converged consisted of six states with seven
pairs of directed edges. Examining the state occupancies during the voltage-clamp protocols
shows that the model possesses two closed states which are occupied at rest, one open
state, an inactivated state with fast recovery, and a transition state that is needed to improve
fits to the voltage activation curve at voltages beyond 0 mV. The model also contains
contains a separate slow inactivated state in which channel occupancy accumulates during
an induction protocol with repeated depolarizing steps. Incorporated this sodium channel
model into an existing CA1 pyramidal neuron model [9] produced
a model that accurately reproduce activity-dependent attenuation of backpropagating action
potentials [10, 11, 12]. Eliminating the prolonged inactivation of the sodium conductance, of
course, abolished the activity dependence. Further simulations also showed that prolonged
inactivation of sodium channels can result in the experimentally-observed inhibition of
dendritic spikes by prior local depolarization, either from backpropagating action potentials
[13] or from prior dendritic spikes [14].
References
[1] A. L. Hodgkin and A. F. Huxley, “A quantitative description of membrane current and its
application to conduction and excitation in nerve,” J. Physiol. 117, 500–544 (1952).
[2] B. Rudy, “Slow inactivation of the sodium conductance in squid giant axons. Pronase
resistance,” J Physiol 283, 1–21 (1978).
[3] I. A. Fleidervish, A. Friedman, and M. J. Gutnick, “Slow inactivation of Na+ current and
slow cumulative spike adaptation in mouse and guinea-pig neocortical neurones in slices,” J
Physiol 493, 83–97 (1996).
[4] H. Y. Jung, T. Mickus, and N. Spruston, “Prolonged sodium channel inactivation
contributes to dendritic action potential attenuation in hippocampal pyramidal neurons,” J.
Neurosci. 17, 6639–6646 (1997).
[5] T. Mickus, H. Jung, and N. Spruston, “Properties of slow, cumulative sodium channel
inactivation in rat hippocampal CA1 pyramidal neurons,” Biophys J 76, 846–860 (1999).
[6] F. P. Boynton, “Detailed Balance + Microscopic Reversibility,” J. Chem. Phys. 40, 3124
(1964).
[7] D. Colquhoun, K. A. Dowsland, M. Beato, and A. J. Plested, “How to impose microscopic
reversibility in complex reaction mechanisms,” Biophys J 86, 3510–3518 (2004).
[8] J. P. Ignizio, Goal Programming and Extensions (Lexington Books, Lexington, MA, 1976).
[9] N. L. Golding, T. J. Mickus, Y. Katz,W. L. Kath, and N. Spruston, “Factors mediating
powerful voltage attenuation along CA1 pyramidal neuron dendrites,” J Physiol 568, 69–82
(2005).
[10] N. Spruston, Y. Schiller, G. Stuart, and B. Sakmann, “Activity-dependent action potential
invasion and calcium influx into hippocampal CA1 dendrites,” Science 268, 297–300 (1995).
[11] J. C. Callaway and W. N. Ross, “Frequency-dependent propagation of sodium action
potentials in dendrites of hippocampal CA1 pyramidal neurons,” J Neurophysiol 74, 1395–
403 (1995).
[12] N. L. Golding,W. L. Kath, and N. Spruston, “Dichotomy of action-potential
backpropagation in CA1 pyramidal neuron dendrites,” J Neurophysiol 86, 2998–3010 (2001).
[13] N. L. Golding and N. Spruston, “Dendritic sodium spikes are variable triggers of axonal
action potentials in hippocampal ca1 pyramidal neurons,” Neuron 21, 1189–1200 (1998).
[14] S. Remy, J. Csicsvari, and H. Beck, “Activity-dependent control of neuronal output by
local and global dendritic spike attenuation,” Neuron 61, 906–16 (2009).
Efrat Katz and Michael Gutnick
The Hebrew University of Jerusalem, Israel
NEOCORTICAL NEURONS POSSESS TWO DISTINCT PERSISTENT SODIUM
CURRENTS WITH DIFFERENT VOLTAGE DEPENDENCE AND DIFFERENT
UNDERLYING MECHANISM OF GENERATION. The central role of voltage-gated Na+ channels in generation and propagation of action potentials (APs) make them key to neuronal excitability. In addition to the well-described, fast-inactivating component of the Na+ current, neocortical neurons also exhibit a slowly inactivating, persistent Na+ current (INaP), that plays a role in determining AP threshold and in synaptic integration. The mechanisms underlying INaP remain unclear. We previously showed in single channel and whole cell recordings that INaP in pyramidal neurons is primarily generated in the axon initial segment (AIS). This was recently confirmed by imaging Na+ influx during prolonged sub-threshold depolarizations as well as during slow voltage ramps. Imaging experiments also suggest the presence of somatic INaP with a right-shifted voltage dependence. In whole-cell voltage clamp at 22oC, voltage ramps reveal two distinct persistent Na+ currents, each with a distinct voltage-dependence; upon warming to 35oC, the distinction between the currents is lost. Simultaneous Na+ imaging reveals a somatic INaP with an I-V relationship that is shifted by about 15 mV in the depolarizing direction as compared to the axonal INaP. The voltage dependence of the Na flux in the AIS and in the soma suggests that the axonal INaP is generated by ¿window current¿, as predicted by the Hodgkin Huxley formalism, whereas the somatic INaP reflects a periodic failure of individual channels to inactivate (¿modal gating¿).
Lyle N. Long
Distinguished Professor of Engineering and Mathematics
The Pennsylvania State University
http://www.personal.psu.edu/lnl
EFFICIENT AND SCALABLE NEURAL NETWORK SIMULATIONS FOR
ENGINEERING APPLICATIONS USING THE HODGKIN-HUXLEY EQUATIONS
Numerous papers in the literature cite the Hodgkin--‐Huxley (HH) equations as being too
expensive to use in neural network simulations, especially for engineering applications. They
thus often use reduced models or the extremely simple leaky integrate and fire (LIF) model.
Many authors also use programming approaches such as Matlab, which is much slower than
C++. This paper will discuss efficient numerical schemes for solving the HH equations that
are 100 times faster than simple methods, and only 8 times more expensive than the best
methods applied to LIF. We will also show some of the drawbacks and common
misunderstandings of the reduced models. The schemes to be described use algorithms
such as the exponential Euler method combined with table lookup schemes for the
coefficients. Numerical stability, time step size issues, and computational cost will also be
discussed. In addition, in large--‐scale neural networks the computational costs are
dominated by synaptic simulations, since there are orders of magnitude more synapses than
neurons. Efficient simulations offer the promise of more detailed neural networks for
engineering applications (e.g. robotics, sensor processing, control, etc.) as well as brain
simulations. Even on a laptop (2.4 GHz with 2 GB memory) we can run 1 million H--‐H
neurons with 100 million synapses for 100,000 time steps in 1 hour (including an efficient
learning algorithm based on STDP). It is also possible to incorporate efficient neuro--‐ and
synaptogenesis, so that the neural network can adapt to changing inputs and the network
can grow automatically. In addition, modern massively parallel supercomputers could
simulate numbers of neuron near those in the human brain.
Recent papers by the author can be found at:
http://www.personal.psu.edu/lnl/papers.html
Stefano Luccioli
Istituto Sistemi Complessi – CNR, Sesto Fiorentino, Italy.
COHERENCE RESONANCES IN THE HODGKIN-HUXLEY MODEL IN THE HIGH-INPUT
REGIME Our aim is to analyze the response of the Hodgkin-Huxley neuronal model subjected to a large number of uncorrelated stochastic excitatory and inhibitory post-synaptic spike trains. The model is examined in the silent dynamical regime below the saddle-node bifurcation. The response of the neuron is examined by considering statistical indicators (interspike-intervals distributions and their first moments) and dynamical indicators (autocorrelation functions). The main result is the coexistence of two different coherence resonances: one occurs at quite low noise and is related to the coherence of subthreshold oscillations around the rest state; the second one (at intermediate noise variance) is associated with the regularization of the sequence of spikes emitted by the neuron.
Mark D. McDonnell, Brett A. Schmerl and Daniel E. Padilla
Computational and Theoretical Neuroscience Laboratory, University of South Australia
SLOPE-BASED SUPRATHRESHOLD STOCHASTIC RESONANCE DUE TO ION-
CHANNEL NOISE IN PHASIC AUDITORY BRAINSTEM NEURON MODELS
Stochastic variability in ion channel conductances, due to random transitions between ion
channel states, can nontrivially affect neural behaviour. However, whether this ion-channel
noise gives rise to noise-enhanced neural processing is unclear, having received little
attention compared with synaptic noise. A recent exception (Ashida and Kubo, 2010) is
simulations of a population of Hodgkin-Huxley models, with a Markov-based model of
stochastic ion channel noise, that exhibited a noise-enhancing effect called suprathreshold
stochastic resonance (SSR).
We have therefore investigated whether other forms of noise enhanced processing, or
“stochastic facilitation” (McDonnell and Ward, 2011) can be observed due to ion-channel
noise models. Specifically, we replaced current noise with channel noise in a detailed
Hodgkin-Huxley like model of phasically firing neurons in the auditory brainstem that exhibits
a noise-enhancing effect called slope-based stochastic resonance (SBSR), i.e. noise
enables firing in response to slowly varying inputs (Gai, Doiron and Rinzel, 2010).
We found that SBSR persists for a broad range of noise levels (determined by membrane
patch areas), and that SSR and SBSR can be combined to form slope-based SSR. This
suggests intrinsic channel noise might be exploited in-vivo to enable phasic population
responses to robustly encode slowly varying signals.
References:
*MD McDonnell, LM Ward, “The benefits of noise in neural systems: bridging theory and
experiment,” Nature Reviews Neuroscience 12: 415-426, 2011.
*G Ashida, M Kubo “Suprathreshold stochastic resonance induced by ion channel
fluctuation,” Physica D 239:327-334, 2010.
*Y Gai, B Doiron, J Rinzel, “Slope-based stochastic resonance: How noise enables phasic
neurons to encode slow signals,” PLoS Computational Biology 6:e1000825, 2010.
Michele Migliore
Institute of Biophysics, National Research Council (Palermo, Italy)
ON THE MECHANISMS UNDERLYING THE DEPOLARIZATION BLOCK IN THE
SPIKING DYNAMICS OF CA1 PYRAMIDAL NEURONS Under sustained input current of increasing strength neurons eventually stop firing, entering a depolarization block. This is a robust effect that is not usually explored in experiments or explicitly implemented or tested in models. However, the range of current strength needed for a depolarization block could be easily reached with a random background activity of only a few hundred excitatory synapses. The depolarization block may thus be an important property of neurons, which should be better characterized in experiments and explicitly taken into account in models at all implementation scales. Here we analyze the spiking dynamics of CA1 pyramidal neuron models using the same set ionic currents, modeled according to the HH formalism, on both an accurate morphological reconstruction and on its reduction to a single-compartment. The results show why HH-like models are able to show this effect, what are the specific ion channel properties and kinetics that are needed to reproduce the experimental findings, and how their interplay can drastically modulate the neuronal dynamics and the input current range leading to depolarization block. We suggest that this can be one of the rate-limiting mechanisms protecting a CA1 neuron from excessive spiking activity.
Recent relevant papers by the author using HH-like channels:
Bianchi D, Marasco A, Limongiello A, Marchetti C, Marie H, Tirozzi B, Migliore M. (2012) On the mechanisms underlying the depolarization block in the spiking dynamics of CA1 pyramidal neurons. J Comput Neurosci. 2012 Feb 5. [Epub ahead of print]
Halnes G, Augustinaite S, Heggelund P, Einevoll GT, Migliore M (2011) A Multi-
Compartment Model for Interneurons in the Dorsal Lateral Geniculate Nucleus PLoS
Comp. Biol. 7:e1002160.
Ascoli GA, Gasparini S, Medinilla V, and Migliore M. Local Control of Post-Inhibitory
Rebound Spiking in CA1 Pyramidal Neuron Dendrites, J. Neurosci., 30: 6422-6433
(2010).
Ferrante M, Migliore M, Ascoli GA. Feed-forward inhibition as a buffer of the neuronal input-
output relation. Proc. Natl. Acad. Sci. USA, 106:18004-18009 (2009).
Mala M. Shah, Michele Migliore, Ignacio Valencia, Edward C. Cooper and David A. Brown,
Functional Significance of Axonal Kv7 channels in Hippocampal Pyramidal Neurons,
Proc. Nat. Acad. Sci. USA, 105:7869-7874 (2008).
Ferrante M, Blackwell KT, Migliore M, Ascoli GA (2008), Computational models of neuronal
biophysics and the characterization of potential neuropharmacological targets,
Current Medicinal Chemistry 15:2456-2471
Lorin Milescu
University of Missouri, Columbia
PLAYING WITH ION CHANNEL MODELS IN REAL NEURONS
The beautiful work of Hodgkin and Huxley has laid down the foundations for modeling the
kinetics of voltage-gated ion channels, and for understanding the mechanisms of action
potential firing in neurons. Since then, much progress has been made in characterizing the
biophysical events associated with voltage-gating, and more complex kinetic models have
been proposed. Real-time computational techniques have recently made it possible to
determine how the kinetic properties of one channel type modulate the firing activity of a
particular neuron.
With some technical improvements and our own implementation of the dynamic clamp
technique, programmed in the QuB software, we were able to test both Hodgkin-Huxley and
state models of Na channel kinetics in ventral raphé and other types of mammalian central
neurons. First, we searched for a kinetic model and a set of parameters that satisfied the
experimental evidence provided by voltage clamp data. We found that Na channels in
pacemaker raphé neurons have slow inactivation properties that cannot be easily explained
by the traditional H-H formalism, but can be reasonably approximated with state models.
Then, we tested both these model types in raphé neurons using dynamic clamp. We were
able to establish that Na channel slow inactivation is partly responsible for the experimentally
observed adaptation in firing frequency and spike shape, in response to excitatory current.
Without slow inactivation, the H-H model generated firing activity that reproduced well the
spike shape but not the adaptation. Furthermore, we could determine that the subthreshold
Na current contributes to pacemaking but in a limited voltage range, together with other
subthreshold currents. All the tested models failed to reproduce the sharper spike onset
observed experimentally. However, using dynamic clamp to add a virtual axon with a high
density of Na channels restored with good approximation the sharp rising.
I can hardly imagine a more rewarding experience than watching a neuron in a brain slice
respond with action potentials to current injected through a patch pipette and generated on a
multiprocessor computer, on the basis of a Na channel model developed by Hodgkin and
Huxley 60 years ago!
Andreas Neef
MPI Dynamics and Self-Organization, Goettingen, Germany
SOMATIC SODIUM CHANNELS ACCOUNT FOR 2ND PHASE OF ACTION POTENTIAL
UPSTROKE IN LAYER 5 PYRAMIDAL CELLS Mechanisms of action potential (AP) generation in neocortical pyramidal cells have been the focus of intense experimental and theoretical research over the last several decades. It has proven very difficult, however, to arrive at a consensus model which can satisfactorily account for all of its features. One of the still unresolved issues is lack of accurate description of Na+ channel kinetics in different neuronal compartments. Here, we measured kinetics of somatic Na+ channels using high temporal resolution (5-10 kHz, -3dB, low pass four-pole Bessel filter) cell-attached recordings from layer 5 pyramidal neurons in neocortical slices. The data were described by fitting different Markov models with differential evolution fit algorithms. The limited speed of voltage steps and the effect of current filtering were accounted for in the fit procedure. Activation kinetics was best described by Markov models with two sequentially activating gates, while inactivation was best described as a process that runs in parallel to activation. The best model described the channel data well enough to allow quantitative prediction of the somatic Na+ current during the somatic spike. To this end the AP waveform recorded in current clamp in the same preparation, was used to drive Na+ channels in the model. The resulting simulated current matched the second phase of the AP upstroke in the phase plot (dV/dt vs V). This is consistent with the long standing idea that somatic Na+ channels are the main current sink during this second phase of the AP upstroke but contribute little to its initial phase. Besides a precise description of sodium channel activation kinetics, these measurements also provide independent lower bounds on the somatic sodium channel density of about 10 channels per micrometer square. Supported by the GIF and the BMBF (BCCN, Goettingen)
Mohammad Ali Neishabouri Imperial College London
EFFECTS OF NAV1.8 CLUSTERING IN SMALL DIAMETER UNMYELINATED
AXONS NaV1.8 is the main channel responsible for the conduction of action potentials in C-type fibres of sensory neurons (Baker, 2005). We have recently discovered that NaV1.8 has a clustered distribution along the axons of unmyelinated small diameter sensory neurons, and shows co-localisation with lipid raft makers. Disruption of lipid rafts in these neurons can lead to abolition of excitability. We investigated the effects of the clustering of sodium channels on the propagation velocity and associated metabolic cost of action potentials in C-type unmyelinated axons using both deterministic and stochastic simulations in Modigliani (A. A. Faisal, Laughlin, & White, 2002). We simulated uniformly distributed and clustered channels (0.2µm long clusters of channels every 3µm) along the unmyelinated fibres (0.1µm diameter). Deterministic simulations did not show any significant difference between the models. When simulated stochastically, clustered channels conducted action potential more slowly (0.106 m/s vs 0.134 m/s), but they significantly reduced the metabolic cost of conduction (3.12e-6 picomol ATP per mm vs 6.91e-6 picomoal ATP per mm). The results are consistent with previous findings about the effects of noise in small diameter axons (A. Faisal, White, & Laughlin, 2005) and emphasize the need for stochastic simulations in small axons.
Uri Nevo
Tel Aviv University, Tel Aviv, Israel
NEURONAL ACTIVITY, BEYOND ELECTRICITY AND CHEMISTRY
Neuronal activity is an electrical and a chemical process. However, observations suggest
that it includes a mechanical dimension. Neuronal excitation was shown to be accompanied
by cellular swelling and heat production (Iwasa et al.). This swelling was minor (~1nm for a
squid giant axon) but was significant and propagated with the action potential. Diffusion
weighted MRI (DWI) demonstrated that enhanced neuronal activity is accompanied by a
drop in the displacement of water molecules (Darquie et al.). However, the link of this
hydrodynamic effect to neuronal activity was never validated due to the overwhelming
influence of hemodynamic and vascular effects (Miller et al.).
We study the hydrodynamic component of neuronal activity using two experimental models
of isolated and electrically active tissues: isolated and electrically active newborn rat spinal
cords and the CNS of the medicinal leech.
Isolated electrically active spinal cords of newborn Sprague Dawley rats (n=8) were
continuously perfused with oxygenated artificial CSF thus maintained viable. Water
displacement was evaluated during three physiological phases: (a) Spontaneous baseline
activity, (b) enhanced activity (induced by serotonin, NMDA and bicuculline) and (c)
reproducible baseline activity (induced by washing out the excitatory solution). During
elevated neuronal activity, the average mobility of water molecules dropped by 18-27%. The
observed changes were significant and repeated in all preparations. Given the nature of the
isolated experimental system, these results suggest that electrical activity affects water
displacement in mammalian neurons (Tirosh and Nevo, Submitted). Further experiments are
currently performed, to uncover the exact biophysical mechanism in the basis of the
dynamics of water displacement.
In a separate experiment we inject inert quantum dots to neurons in isolated ganglions of the
CNS of the medicinal leech, and track the propagation-front of the quantum dots. Our
preliminary results suggest that colloidal particles propagate within the branches of active
neurons significantly faster, relative to fixed neurons. Further experiments are currently
performed to study the exact propagation mechanism and its relation to neuronal activity.
To conclude, we suggest that neuronal activity includes a mechanical component that is yet
to be understood.
References
Iwasa, K., Tasaki, I. & Gibbons, R. C. Swelling of nerve fibers associated with action
potentials. Science 210: 338-339, 1980.
Darquie A., Poline J. B., Poupon C., Saint-Jalmes H., & Le Bihan D. Transient decrease in
water diffusion observed in human occipital cortex during visual stimulation. P Natl Acad Sci
USA 98(16):9391-9395.
Miller K. L. et al. Evidence for a vascular contribution to diffusion FMRI at high b value. P
Natl Acad Sci USA 104(52):20967-20972.
Tirosh N. & Nevo U. Neuronal activity significantly reduces water displacement: DWI of a
vital rat spinal cord with no hemodynamic effect. Submitted for publication.
Netanel Ofer Bar Ilan University, Israel
THE AXONAL RESPONSE TO CURRENT STIMULI AND MORPHOLOGICAL
MODIFICATIONS - NON-LINEAR ANALYSIS OF THE HODGKIN-HUXLEY MODEL The electric response of a neuron to a constant applied depolarizing current is an interesting phenomenon. Studying this response is important to the understanding of information coding in the nervous system. In the current study, we examine the response of neuronal cells to current stimuli. We use the full Hodgkin Huxley model to investigate in details the axonal response to varied current stimuli. We stimulate an axon with graded levels of continuous DC currents, and examine the response for the space clamped and for the cable propagating models. In addition, we modify the radius of the axon and explore the correlated response. As in previous studies, we find that a low current injection, right above threshold, generates a single action potential. Injection of a higher current generates an infinite train of action potentials with a frequency that correlates with the current amplitude. Increasing further the current reduces the number of action potentials back to a single spike. In our simulation we study the response of an axon to injected currents in between these 'single-spike' and 'infinite-train' regimes and demonstrate two borderline cases. Increasing the current above the 'single-spike' regime leads to a finite number of action potentials. Thus, by changing accurately the stimulating current we can control the train length. The second borderline phenomenon occurs when the injected current increases above the 'infinite-train' regime and causes 'failure' or a delay in firing following a finite series of action potentials. Thus, modifications of the injected currents lead to an additional mechanism to control spikes series lengths. Using non-linear dynamics analysis we explore the fundamental mechanisms that lead to the presented borderline phenomena. The later together with effects of morphological modifications on the pattern of activity will be discussed and interpreted as instruments for information coding. Netanel Ofer and Orit Shefi Faculty of Engineering and Bar Ilan Institute of Nanotechnology and Advanced Materials, Bar Ilan University, 52900, Ramat Gan.
Wilfrid Rall and Gordon M. Shepherd *
retired NIH, Bethesda, * Yale University
FROM HODGKIN & HUXLEY TO THE CENTRAL NERVOUS SYSTEM: FIRST STEPS
IN BUILDING BIOPHYSICALLY-REALISTIC EXCITABILITY INTO CENTRAL NEURONS
AND THEIR DENDRITES.
The Hodgkin-Huxley action potential model for the squid axon in 1952 implied a similar
mechanism for all neurons, but demonstrating this required overcoming two main problems.
First was how to represent the synaptic integration within the complex branching of neuronal
dendrites that underlies the control of action potential initiation in the soma and axon hillock.
Building on cable properties, compartmental modeling of the soma and branching dendritic
systems on the digital computer was developed (Rall, 1964). This was first applied to gain
insight into the integrative actions of synaptic excitatory and inhibitory conductances and
potentials in spinal motor neuron dendrites. The second main problem was the limited
computing resources of that time for representing both dendritic complexity and the HH
equations. Based on the insights of Hodgkin, Huxley and Katz, we made use of an abstract
membrane equivalent circuit with excitatory and inhibitory conductance pathways. To
facilitate action potential computations we approximated the exponential nonlinearity of the
HH model with polynomial nonlinearity.
These developments came together in a study in the mammalian olfactory bulb of antidromic
activation of mitral cells followed by synaptic inhibition. To test the postulate that the
inhibition was due to a synaptic pathway through granule cell interneurons (Shepherd,
1963), a multidisciplinary project was carried out combining extracellular and intracellular
recordings, field potential recordings, anatomical correlations, and the first computer
modeling to simulate such physiological activity. The model simulated antidromic action
potential invasion of the mitral cell axon, cell body and dendrites, followed by synaptic
excitation of the granule cell (Rall & Shepherd, 1968). Detailed comparison with
experimental data led to the postulate of a dendrodendritic synaptic pathway between the
mitral cell dendrites and granule cell spines. The postulate was supported by the EM finding
of side by side excitatory mitral to granule synapses and inhibitory granule to mitral
synapses (Rall et al, 1966).
This study showed a function for an action potential spreading centrifugally
(backpropagating) from the soma; the geometrical hurdle in invading from the thin axon to
the thick soma-dendrites; comparison between active and passive invasion of the dendrites;
reconstruction of field potentials around the synchronously activated mitral and granule cell
populations which helped to localize the sites of synaptic interactions; possible effects of the
field potentials on local membrane potentials; evidence for presynaptic dendrites; and the
fact that dendrites are not limited to receptive functions, as posed by the classical neuron
doctrine, but have synaptic outputs as well.
The simplified HH model was also used to analyse transmission at sites of diameter change
(Goldstein & Rall, 1974). Subsequent studies provided the first models combining full HH
equations with compartmental modeling of mammalian spinal motoneurons (Dodge &
Cooley, 1973), Renshaw cells (Traub, 1977) and hippocampal pyramidal neurons (Traub &
Llinas, 1979). Thresholding of activity in dendritic spines (Perkel & Perkel, 1985; Miller et al,
1985; Rall & Segev, 1987) and spine logic operations (Shepherd & Brayton, 1987) were
explored. With the advent of the personal computer, increasingly comprehensive neuronal
and circuit models have become possible, and modifications of the HH model for mammalian
neurons are being developed, as will be detailed at this conference. The models will always
push the limits of computing speed and capacity, making lessons from the early years still
relevant (Segev & Rall, 1998).
References Rall W (1964) Theoretical significance of dendritic trees for neuronal input-output relations. In Neural Theory and Modelling (Reiss RF ed). Palo Alto: Stanford University Press, pp. 73-97. Shepherd GM (1963) Neuronal systems controlling mitral cell excitability. J Physiol (Lond) 168:101-117 Rall, W, Shepherd GM (l968) Theoretical reconstruction of field potentials and dendro-dendritic synaptic interactions in olfactory bulb. J. Neurophysiol. 3l:884-9l5. Rall W, Shepherd GM, Reese TS, Brightman MW (l966) Dendro-dendritic synaptic pathway for inhibition in the olfactory bulb. Exp. Neurol. l4:44-56 Goldstein SS, Rall W (1974) Changes of action potential shape and velocity for changing core conductor geometry. Biophys J 10:731-757 Dodge FA Jr, Cooley JW (1973) Action potential of the motoneuron. IBM J Res Div 17:219-229. Traub RD (1977) Repetitive firing of Renshaw spinal interneurons. Biol Cybern. 2:71-76. Traub RD, Llinas R (1979) Hippocampal pyramidal cells: significance of dendritic ionic conductances for neuronal function and epileptogenesis. J Neurophysiol 42:476-498. Miller JP, Rall W, Rinzel J (1985) Synaptic amplification by active membrane in dendritic spines. Brain Res 325:323-330 Rall W, Segev I (1987) Functional possibilities for synapses on dendrites and on dendritic spines. In Synaptic Function (Edelman GM, Gall WF, Cowan WM eds). New York: John Wiley & Sons, pp. 605-636. Shepherd, G.M. and R.K. Brayton (1987) Logic operations are properties of computer-simulated interactions between excitable dendritic spines. Neurosci. 21:151-166. Segev I, Rall W. (1998) Excitable dendrites and spines: earlier theoretical insights elucidate recent direct observations. Trends Neurosci. 11:453-60.
Indira M. Raman
Northwestern University
RESURGENT CURRENT OF VOLTAGE-GATED NA CHANNELS
The voltage-gated, TTX-sensitive Na channels of Purkinje cells and several other cerebellar
neurons open upon depolarization and are rapidly blocked by an endogenous protein that
competes with the classical fast inactivation gate. Upon repolarization, permeating Na ions
rapidly expel the open-channel blocker, producing a “resurgent” Na current. The cycle of
opening and blocking upon depolarization and reopening and unblocking upon repolarization
prevents channels from accumulating in long-lived inactivated states during spiking, thereby
reducing refractory periods and promoting rapid firing. Resurgent current has now been
found in 15 cell types, many of them typified by rapid firing or bursting.
Investigating the molecular basis of resurgent current, we find that open-channel block is
most effective on the Na channel subunit NaV1.6 in Purkinje cells, because those channels
enter fast inactivated states relatively slowly, giving the blocker time to bind. The blocking
protein associates with other subunits but cannot always produce resurgent current;
resurgent current can be revealed in these channels by pharmacologically slowing
inactivation. The subunit NaV4 (scn4b) has emerged as a likely candidate for the blocking
protein, based on the sequence of its cytoplasmic tail, which resembles (non-native) open-
channel blocking peptides. Indeed, a free peptide replicating the 4 tail induces resurgent
current in channels lacking an open-channel blocker, and knockdown of NaV4 with siRNA in
cerebellar granule cell cultures reduces or abolishes resurgent current and disrupts regular
firing. The efficacy of block depends on a FxxKK sequence in the NaV4 cytoplasmic tail.
Our ongoing work indicates that both the native blocker and the 4 peptide antagonize Na
channel inhibition by lidocaine, possibly by competing for an overlapping binding site.
Interestingly, lidocaine and other use-dependent blockers are used clinically to prevent
excessive firing associated with pain and epilepsy; our data predict that cells with resurgent
current will be relatively resistant to inhibition by these drugs, possibly protecting cells with
naturally rapid firing patterns from clinical agents that suppress such activity.
Hugh Robinson
PDN, University of Cambridge
MECHANISMS AND FUNCTIONS OF IRREGULAR SPIKING IN A CLASS OF
NEOCORTICAL INHIBITORY INTERNEURONS
One of the organizing principles of electrical activity in the cortex is the appearance of
distinct synchronous oscillations at various frequencies from slow (< 1 Hz) to very fast (> 100
Hz), for example the theta (4-10 Hz), beta (10-30 Hz) and gamma (30 – 80 Hz) oscillations,
which are implicated in cognition, attention, and sensorimotor processing, and may subserve
encoding and processing of information through spike timing. Within the local neural circuitry
of the neocortex are various types of inhibitory interneuron, several of which have been
shown to form specific gap-junction coupled networks allowing them to fire as synchronous
units, and they appear to play important roles in generating these oscillations.
Oscillations require a combination of periodic firing in individual cells and synchrony. The
majority of excitatory, pyramidal neurons in the cortex, so-called regular-spiking cells, fire
regularly in response to a constant stimulus. Most types of interneuron do likewise,
sometimes following adapting or anti-adapting initial transients, or they fire in regular or
irregular bursts of periodic firing. One type of interneuron, however, is distinguished by its
intrinsically irregular repetitive firing, showing a broad, apparently random dispersion of its
spike intervals, even when pharmacologically disconnected from any synaptic input. These
irregular-spiking (IS) neurons must have both a distinctive mechanism of spike generation,
and an unusual role during synchronous network oscillations. To enable specific targeting of
this interneuron type, we used a mouse line in which GFP is linked to the promoter for
GAD65, in which fluorescently labelled neurons in somatosensory cortex have an IS
phenotype. It has been shown that these cells are CCK-expressing, are concentrated in a
band in layer 2, and connect specifically to each other by gap junctions and mutually
inhibitory synaptic connections which together enable precisely-synchronized irregular firing.
Their wide axonal arborizations through many layers of the cortex and inhibition of pyramidal
cells suggest that they could exert a powerful influence on the network. Although they make
up a large proportion of inhibitory interneurons in upper layer 2/3, they have received much
less attention than other classes of interneurons, such as FS and LTS cells.
In this study, we ask: what mechanisms do these neurons use to achieve their striking
irregularity of firing, and what are the functional consequences of this in the oscillating
cortical network? The basis of regular repetitive firing of action potentials by neurons is well-
understood in simple biophysically-based dynamical models. With a constant level of
stimulus, action potentials are produced as the system becomes attracted to a “limit cycle” in
the dynamical phase space. Intrinsically irregular spike timing, however, is more difficult to
understand. Some irregularity must arise simply from the stochastic nature of the gating of
ion channels. However, the nonlinear dynamics of the voltage-dependent ion channels
involved in spike generation could also contribute to irregular patterns of membrane
potential, in what may be termed active irregularity.
Here, using a combination of patch-clamp recording in slices of somatosensory cortex, time
series analysis and computational modelling, we show that IS neurons generate robust,
intrinsically irregular firing using both of these strategies. Nonlinear interactions of voltage-
dependent currents give rise to predictability of interspike intervals. The degree of irregularity
is tuned by the level of a voltage-dependent potassium conductance, and interacting
voltage-dependent sodium and potassium channel openings contribute a high level of
voltage-dependent single-channel noise at threshold. The effect of these mechanisms is that
these cells strongly reject synchronization to network frequencies below about 20 Hz, while
synchronizing effectively to higher, gamma frequencies, a property which could give them a
prominent role in gating local cortical gamma oscillations.
Joint work with Ole Paulsen and Mariana Vargas-Caballero.
Frederic Roemschied Bernstein Centre for Computational Neuroscience, Berlin, Germany
TRANSDUCTION EFFECTS ON TEMPERATURE-INVARIANCE OF GRASSHOPPER
AUDITORY RECEPTOR NEURON SPIKE RATES Ion channel dynamics are known to depend on temperature. Therefore, changes in the ambient temperature are likely to affect single-neuron activity in poikilothermic invertebrates, unless temperature-compensation is present at the cellular or network level. We investigated how responses of grasshopper auditory receptor neurons to auditory noise pulses of varied intensity were affected by shifts in temperature. We found that temperature shifts left the entire stimulus-response curve remarkably unaffected. The receptor neuron response is shaped by two consecutive underlying cell-intrinsic processes: stimulus transduction and spike generation, each of which can be assumed to be temperature-dependent. Moreover, no network input is known to exist for the receptor neurons, such that the observed temperature-invariant responses are likely to arise from neuron-intrinsic properties. While spike-generation itself can already be temperature-invariant (cp. talk by S. Schreiber), we here investigated specifically how a combination of temperature-dependent transduction and spike-generation can lead to temperature-invariant stimulus-response curves in a conductance-based neuron model, in absence of network effects. Our results suggest that a temperature-dependent transduction can be beneficial for temperature-invariance of the stimulus-response curve, as it can extend the range of spike-generation processes associated with temperature-invariance.
Jan-Hendrik Schleimer* and Martin StemmlerƗ
*Bernstein Center for Computational Neuroscience, Humboldt-Universität zu Berlin,
Philippstr. 13, Haus 6, 10115 Berlin, Germany
ƗBernstein Center for Computational Neuroscience, Ludwig-Maximilians-Universität
München, Grosshadernerstr. 2, 82152 Planegg, Germany
PHASE REDUCTION OF NOISY HODGKIN-HUXLEY MODELS
Markov chain models for the microscopic ion channel kinetics can mirror the behavior of the
four-dimensional Hodgkin-Huxley model of the squid giant axon [1]. The ion channel noise,
however, resides in a 12-dimensional state space [2], inducing nontrivial temporal
correlations in the noise. Nonetheless, we show that a systematic reduction to a stochastic
differential equation based on a single phase is feasible. Such a reduced system captures
the variance of interspike intervals and the spike jitter that affects neural coding and
computation.
The phase description also allows us to predict the filtering properties of a neuron, which are
a function both of the noise and the dynamics [3]. As a conductance-based model is driven
from quiescence to tonic spiking, the nature of the transition to spiking can produce notch
and bandpass filters, depending on whether the bifurcation is homoclinic, Hopf, or saddle-
node. In this way, the coding properties of individual neurons can be related to the properties
of ion channels embedded in the membrane.
[1] F. Fox, Y. Lu. Phys. Rev. E, 49(4):3421–3431, 1994.
[2] J. Goldwyn, N. Imennov, M. Famulare, E. Shea-Brown. Phys. Rev. E, 83:041908, 2011.
[3] J.-H. Schleimer, M. Stemmler. Phys. Rev. Lett., 103:248105, 2009.
Susanne Schreiber
In collaboration with Frederic Roemschied, Monika Eberhard, and Bernhard Ronacher
Institute for Biology, Humboldt-Universität zu Berlin and Bernstein Center for Computational
Neuroscience Berlin, Germany
GRASSHOPPER NEURONS KEEP COOL IN HOT SITUATIONS: TEMPERATURE-
COMPENSATED SPIKE RATES CAN BE ACHIEVED IN SINGLE NEURONS AT NO
ADDITIONAL ENERGETIC COST
While mammals and birds have evolved an elaborate system to keep their body temperature
constant, body temperature in invertebrates is subject to changes in the surrounding world.
The nervous system of invertebrates hence faces the challenge of preserving its functionality
while its neuronal dynamics are sped up or slowed down depending on the temperature of
the environment. Grasshoppers, for example, have to reliably recognize details of the
temporal structure of a song of a potential mate under conditions that can differ by 10
degrees Celsius or more. It may hence be a useful evolutionary strategy to reduce the
temperature dependence of neuronal responses that play a role in the recognition of these
signals. We investigated this hypothesis based on electrophysiological recordings in the
locust auditory periphery and used conductance-based modeling to identify how robustness
of the neuronal firing rate to temperature changes can be mediated by ionic conductances.
Responses of receptor neurons to acoustic stimulation revealed a low dependence of
neuronal firing rate on temperature in the behaviorally relevant range. This result is
surprising, as receptor neurons are known to constitute the first layer of a simple feed-
forward network and are not assumed to receive network input. The observed temperature
robustness is hence likely to result from cell-intrinsic mechanisms. To shed light on this
finding we analyzed how temperature invariance of spike rate can be achieved in a well-
known conductance-based neuron model by Connor et al. (1977), assuming an influence of
temperature on the dynamics of ion channels in terms of transition rates, peak
conductances, and reversal potentials. Based on this analysis, we propose ionic
mechanisms that underlie temperature invariance of the spike rate. Another important
evolutionary constraint for animals with a large surface-to-volume ratio is energy. We hence
further explored how temperature invariance relates to energy efficiency; in other words we
asked, whether temperature invariance is an energetically-expensive feature. We found that
the energetic cost of an action potential is determined by largely distinct factors: while
temperature dependence of the A-type potassium conductance is a key player in setting
temperature invariance of the spike rate, energy consumption was influenced most by the
temperature-dependence of the sodium inactivation. Overall, our analysis shows that cell-
intrinsic mechanisms are well suited to achieve temperature robustness of neuronal
dynamics at no additional metabolic cost.
Recent relevant papers by the author
1. Clemens J, Kutzki O, Ronacher B, Schreiber S, and Wohlgemuth S (2011). Efficient
transformation of an auditory population code in a small sensory system. PNAS 108
(33):13812-13817; doi:10.1073/pnas.1104506108
2. Schreiber S, Machens CK, Herz AVM, and Laughlin SB (2002). Energy-efficient coding
with discrete stochastic events. Neural Comput. 14(6):1323-46.
Idan Segev
Department of Neurobiology, The Interdisciplinary Center for Neural Computation and the
Edmond and Lily Safra Center for Brain Sciences, the Hebrew University, Jerusalem, Israel.
FROM A SINGLE H&H SPIKE TO A FAMILY OF SPIKE PATTERNS
Neurons, in particular cortical inhibitory interneurons, exhibit a rich repertoire of spike firing.
Each such electrical class (e-class) has a unique electrical fingerprint – an action potential
with a particular shape, as well as a particular pattern of spike firing in response to
depolarizing currents steps. Within a given e-class, a large variability (“noise”) in spike firing
is observed. This electrical diversity of cortical neurons is believed to play a key role in
determining the dynamics of the cortical network. We present a novel framework for
automatically constraining parameters of Hodgkin and Huxley (H&H) - based neuron models,
given a large set of experimentally measured responses of these neurons. Because it was
shown experimentally that intrinsic membrane noise gives rise to a large variability in the
neuron’s response, we argue that the common approach of fitting the model by direct
comparison to a single experimental target trace using one error function should be replaced
by a multiple objective optimization (MOO) approach that allows to use several error
functions jointly. We demonstrate the success of this approach, when used in conjunction
with genetic algorithm optimization, in generating an excellent fit between model behavior
and the firing pattern of essentially all known e-types of cortical neurons. Furthermore, using
the MOO approach, we propose a novel framework inspired by learning theory, for
objectively selecting the stimuli that best unravel the neuron’s spike dynamics. The efficacy
of stimuli is assessed in terms of their ability to constrain the parameter space of
biophysically H&H-based models that faithfully replicate the neuron’s dynamics as attested
by their ability to generalize well to the neuron’s response to novel experimental stimuli. This
general framework paves a new way for defining, evaluating and standardizing effective
electrical probing of neurons, laying the foundation for a much deeper understanding of the
electrical nature of these highly sophisticated and non-linear devices and of the neuronal
networks that they compose.
Biswa Senguptaa S.B. Laughlinb J.E. Nivenc M. Stemmlerd
a Centre for Neuroscience, Indian Institute of Science, India b Neural Circuits Design Group, University of Cambridge, UK c School of Life Sciences, University of Sussex, UK d Bernstein Center for Computational Neuroscience, LMU Munich, Germany
ENERGETICALLY OPTIMAL ACTION POTENTIALS
The design of engineering and biological systems is influenced by the balance between the
costs incurred and the benefits realised. In stark contrast to man-made systems, it is only
recently that we have been able to reverse engineer biological systems to analyse the trade-
offs incurred during natural selection (Alexander, 1996; Sutherland, 2005). One such
example is the nervous system, whose design is subject to metabolic energy constraints
(Niven & Laughlin, 2008).
A large proportion of neurones in the nervous system display pulsatile spiking behaviour,
represented mathematically by relaxation oscillations. Such oscillations emerge from
opposing currents owing on at least two different time scales, fast and slow. Previous work
has quantified the energy cost of these currents in different models of neurones, concluding
that experimentally measured biophysical parameters are locally optimal (Sengupta,
Stemmler, Laughlin, & Niven, 2010). In this talk, we address whether a globally optimal
action potential (AP) exists that uses the least possible energy. For this purpose, we devise
a novel framework for studying optimality in general nonlinear, periodic dynamical systems
by combining adjoint methods of sensitivity analysis with multiple-scales and Poincaré-
Lindstedt perturbation theory. With this technique, we can optimise any functional, such as
the energy, the voltage variance, or even the information rate carried in the spike times
based on time-varying input stimuli, and handle thousands of model parameters.
We show that an AP can be produced without voltage-sensitive K+ channels, but these are
required to make fast APs. Four additional properties make fast APs energy efficient - Na+
channel inactivation, voltage-dependent channel kinetics, co-operativity in K+ channels as
classically described by multiple gating particles, and high channel densities. Together they
implement bang-bang control by producing two brief, non-overlapping pulses of current, and
achieve 98% efficiency. Inwardly-rectifying K+ channels further enhance efficiency by
reducing a major constraint, the load resistance. Thus, energy-efficient signalling pulses are
readily produced by a small set of non-linear mechanisms that, like many signalling proteins,
can be finely tuned.
References
Alexander, R. M. (1996). Optima for animals. Princeton University Press.
Niven, J. E., & Laughlin, S. B. (2008, Jun). Energy limitation as a selective pressure on the
evolution of sensory systems. J Exp Biol, 211 (Pt 11), 1792{1804.
Sengupta, B., Stemmler, M., Laughlin, S. B., & Niven, J. E. (2010). Action potential energy
efficiency varies among neuron types in vertebrates and invertebrates. PLoS Comput Biol, 6
, e1000840.
Sutherland, W. J. (2005, Jun). The best solution. Nature, 435 (7042), 569.
Joshua Singer
University of Maryland
REPETITIVE SPIKING IN AN AXONLESS RETINAL NEURON Several types of retinal interneurons exhibit spikes but lack axons. One such neuron is the AII amacrine cell, in which spikes recorded at the soma exhibit small amplitudes (<10 mV) and broad time courses (>5 ms). Here, we used electrophysiological recordings and computational analysis to examine the mechanisms underlying this atypical spiking. We found that somatic spikes likely represent large, brief action potential-like events initiated in a single, electrotonically distal dendritic compartment. In this same compartment, spiking undergoes slow modulation, likely by an M-type K conductance. The structural correlate of this compartment is a thin neurite that extends from the primary dendritic tree: local application of TTX to this neurite, or excision of it, eliminates spiking. Thus, the physiology of the axon- less AII is much more complex than would be anticipated from morphological descriptions and somatic recordings; in particular, the AII possesses a single dendritic structure that controls its firing pattern.
David Sterratt
University of Edinburgh
A VERY FINE SPATIOTEMPORAL GRID IS REQUIRED TO DETERMINE
PROPAGATION SPEED ACCURATELY IN A MODEL MYELINATED NERVE In simulations of the myelinated nerve model of McIntyre et al. (J. Neurophys. 87:995-1006, 2002) implemented in NEURON (http://senselab.med.yale.edu/ModelDb/showmodel.asp?model=3810) the spatial and temporal discretisations are large relative to the length and membrane time constants. For some parameter values in the model, decreasing the maximum compartment size from 0.1 to 0.0001 of the length constant decreases the simulated propagation speed by around 20%. Decreasing the time constant from 0.0025ms to 10-5 ms increases the propagation speed by around 5%. It is possible to rescue the fit to data by altering the maximum conductance of the sodium and potassium currents, which are free parameters in the model.
Nobuyuki Takahashi
Hokkaido University of Education, Hakodate, Japan
SYNCHRONIZATION OF NERVE IMPULSES COUPLED WITH ELECTROSTATIC
CAPACITANCE IN PARALLEL AXONS MODEL A ladder-like circuit consisting of tandem repeats of Hodgkin-Huxley model was analyzed computational. Synchronization of action potentials was simulated depending on the coupling with electronic capacitance between the tandem repeats of Hodgkin-Huxley model. A role of the synchronization of action potentials on making rhythms in neural systems was discussed.
Yulia Timofeeva
University of Warwick, UK
INTEGRATED NEURAL MODELLING OF CALCIUM AND ELECTRICAL SIGNALLING
Calcium is known to be actively involved in regulating a large variety of neuronal processes including excitability, associativity, neurotransmitter release and synaptic plasticity. The entry of Ca2+ from the extracellular medium into the cell is primarily regulated by voltage-gated calcium channels (VGCCs). Another important contribution to Ca2+ concentration is provided by the Ca2+ release from the endoplasmic reticulum through the inositol (1,4,5)-triphosphate IP3 receptors and the ryanodine receptors. Here we will present a biophysically realistic and computationally inexpensive model of a nerve cell that can incorporate the notions of either discrete or continuous distributions of VGCCs and calcium-sensitive receptors. The proposed model can be solved by using a mixture of analysis and numerical simulations and it provides a better understanding of the interaction between membrane voltage and action potential-evoked Ca2+ signals.
Alessandro Torcini
CNR, Florence, Italy
COHERENT RESPONSE OF THE HODGKIN-HUXLEY MODEL IN THE HIGH-INPUT
REGIME
The response of the classical Hodgkin-Huxley (HH) neuronal model subjected to stochastic
uncorrelated spike trains originating from a large number of inhibitory and excitatory post-
synaptic potentials is analyzed in detail [1,2]. The model is examined in its three fundamental
dynamical regimes: silence, bistability and repetitive firing. Its response is characterized in
terms of statistical indicators (interspike-interval distributions and their first moments) as well
as of dynamical indicators (autocorrelation functions and conditional entropies). In the silent
regime, the coexistence of two different coherence resonances is revealed: one occurs at
quite low noise and is related to the stimulation of subthreshold oscillations around the rest
state; the second one (at intermediate noise variance) is associated with the regularization of
the sequence of spikes emitted by the neuron. Bistability in the low noise limit can be
interpreted in terms of jumping processes across barriers activated by stochastic
fluctuations. In the repetitive firing regime a maximization of incoherence is observed
at finite noise variance. The mechanisms responsible for the different features appearing in
the interspike-interval distributions (like multimodality and exponential tails) are clearly
identified in the various regimes. As a final aspect we analyze the influence of correlations
among the excitatory and inhibitory spike trains on the responsiveness of the HH model. In
particular, the regularity of the emitted spike trains becomes maximal in correspondence to
an optimal correlation among excitatory (or inhibitory) inputs for a fixed mean and variance
of the stochastic input. We expect that a maximum coherence in the neuron response can
be found for an optimal combination of noise level and correlation, similarly to what shown
for the FitzHugh-Nagumo model [3].
Related References
1) S. Luccioli, T. Kreuz, and A. Torcini "Dynamical response of the Hodgkin-Huxley model in
the high-input regime", Phys. Rev. E 73 (2006) 041902
2) A. Torcini, S. Luccioli, and T. Kreuz "Coherent response of the Hodgkin-Huxley neuron in
the high-input regime", Neurocomputing 70 (2007) 1943
3) T. Kreuz, S. Luccioli, and A. Torcini, "Double coherence resonance in neuron models
driven by discrete correlated noise", Phys. Rev. Lett. 97 (2006) 238101
More details about my profile and a list of all my publications
can be found here:
http://www.fi.isc.cnr.it/users/alessandro.torcini/
Henry C. Tuckwell* and Jürgen Jost
Max Planck Institute for Mathematics in the Sciences
Inselstr 22, 04103 Leipzig, Germany
INVERSE STOCHASTIC RESONANCE AND LONG-TERM FIRING PROPERTIES IN
STOCHASTIC HODGKIN-HUXLEY SYSTEMS
When a Hodgkin-Huxley model neuron is subject to a current with a steady mean µ of
certain values, repetitive firing ensues as long as the current is maintained. Noise can
dramatically change the firing pattern, most noticeably for some values stopping the firing
completely and indefinitely. This has been confirmed in experiments on squid axons.
Increasing the noise level can lead to a minimum in the firing rate, which is referred to as
inverse stochastic resonance. This has been demonstrated in the Hodgkin-Huxley systems
with and without spatial extent and with various initial data and input current types. In the
spatial model it was found that weak noise could strongly inhibit the instigation of firing but
had little effect on the propagation of action potentials. Recently, we have analysed the
effects of noise in the long term firing behaviour of the Hodgkin-Huxley space clamped
system by examining the underlying transitions from the basins of attraction of the stable
equilibrium point to that of the limit cycle and vice-versa. Focusing on the case of a mean input current density at which repetitive firing occurs and ISR had been found to be pronounced, some of the properties of the corresponding stable equilibrium point are found. A linearized approximation around this point has oscillatory solutions from whose maxima spikes tend to occur. A one dimensional diffusion is also constructed for small noise based on the correlations between the pairs of HH variables and the small magnitudes of the fluctuations in two of them. Properties of the basin of attraction of the limit cycle (spike) are investigated and also the nature of distribution of spikes at very small noise corresponding to trajectories which do not ever enter the basin of attraction of the equilibrium point. Long term trials of duration 500000 ms are carried out for values of the noise parameter σ from 0 to 2.0. The graph of mean spike count versus σ is divided into 4 regions R1,...,R4, where R3 contains the minimum associated with ISR. In R1 noise has practically no effect until a critical value of σ1 is reached. At a larger critical value σ2, the probability of transitions from the basin of attraction of the equilibrium point to that of the limit cycle becomes greater than zero and the spike rate thereafter increases with increasing σ. The quantitative scheme underlying ISR is outlined in terms of exit time random variables and illustrated diagrammatically. Several statistical properties of the main random variables associated with long term spiking activity are given, including distributions of exit times from the two relevant basins of attraction and the interspike interval. Application of these ideas a to model of pacemaking in a serotonergic neuron of the dorsal raphe nucleus is also discussed.
Recent relevant papers;
H.C. Tuckwell, J.Jost, B.S. Gutkin, Inhibition and modulation of rhythmic neuronal spiking by
noise. Phys. Rev. E 80 (2009) 031907.
H.C. Tuckwell, J. Jost J, Moment analysis of the Hodgkin-Huxley system with additive noise.
Physica A 388 (2009) 4115-4125.
H.C.Tuckwell, J. Jost, Weak noise powerfully inhibits rhythmic spiking but not its
propagation. PLoS Comp. Biol. 6 (2010) e1000794.
H.C. Tuckwell, J. Jost, The effects of various spatial distributions of weak noise on rhythmic
spiking. J. Comp. Neurosci. 30 (2011) 361-371.
H.C. Tuckwell, J. Jost , Analysis of inverse stochastic resonance and the long-term firing of
Hodgkin-Huxley neurons with Gaussian white noise, submitted for publication (2012).
Maxim Volgushev
University of Connecticut
FAST COMPUTATIONS IN CORTICAL ENSEMBLES REQUIRE RAPID INITIATION OF
ACTION POTENTIALS The abilities of neuronal populations to encode rapidly-varying stimuli and respond fast to abrupt input changes are crucial for basic neuronal computations, such as coincidence detection, grouping by synchrony and spike-timing-dependent plasticity, as well as for the processing speed of neuronal networks. Theoretical analysis has linked these abilities to onset dynamics of action potentials (APs). Using a combination of whole cell recordings from neurons in rat neocortex and computer simulations, we provide first experimental evidence for this conjecture and prove its validity for the case of distal AP initiation in the axon initial segment (AIS), typical for cortical neurons. We show that neocortical neurons generating APs with fast onset dynamics can encode rapidly changing signals by phase-locking their population firing to signal frequencies up to ~300-400 Hz and respond very fast, within 1-2 ms, to subtle changes of input current. In multicompartment conductance based models that reproduce spatial pattern of AP initiation in the AIS, encoding depends on AP onset dynamics at the initiation site. In neocortical neurons in which AP onset was slowed down by experimental manipulations, or was intrinsically slow due to immature AP generation mechanisms, the ability to encode high frequencies and response speed were dramatically reduced. We conclude that fast onset dynamics is a genuine property of cortical AP generators. It enables fast computations in cortical circuits, rich of recurrent connections both within each region and across the hierarchy of areas.
Fred Wolf
Max Planck Institute for Dynamics and Self‐Organization, Göttingen, Germany; Bernstein
Center for Computational Neuroscience Göttingen, Germany; DFG Research Center
889„Cellular Mechanisms of Sensory Processing“ Göttingen, Germany
HOW DETAILS MATTER ‐ COMPUTATIONAL CAPABILITIES OF NEOCORTICAL
NETWORKS REFLECT THE PHYSIOLOGY OF ACTION POTENTIAL INITIATION
Over the past 20 years, theories of neocortical network computation and dynamics rapidly
developed, based on the simplifying assumption that details of single neuron operation and
in particular the detailed biophysics of action potential initiation could be safely ignored.
Currently, this picture and approach is rapidly changing – giving way to an increasingly
accurate understanding of how fine details of neuronal action potentials shape speed,
coordination, and information representation in networks of neocortical networks. My talk will
revisit key steps that led network theorists to realize the importance and impact of action
potential properties for collective network computation. It will revisit failed and successful
attempts to base the analysis of collective cortical dynamics on a Hodgkin-Huxley model
description of single neuron action potential initiation. It will furthermore highlight how
mathematical concepts developed for abstract theoretical purposes are currently catalyzing
a new generation of studies that quantitatively dissect action potential generation and
encoding in biological neurons embedded in emulated, simulated, and real cortical networks.
I will argue that effects such as the submillisecond speed of firing rate changes in cortical
populations (Tchumatchenko et al. J Neurosci 2011) or the high rates of information
generation and decay in cortical networks (Monteforte & Wolf Phys Rev Lett 2010 ) recently
described by my group cannot be appropriately understood without a solid mechanistic
linkage to fine details of action potential generation. Viewed together these developments
indicate that we are only starting to appreciate the power and intricacy with which the
elementary processes of action potential generation‐first formulated with mathematical rigor
and precision by Alan Lloyd Hodgkin und Andrew Fielding Huxley‐ impact on the
computational capabilities and collective dynamics of neuronal systems.
Recent relevant papers
1. B. Naundorf, T. Geisel, and F. Wolf. Action potential onset dynamics and the response
speed of neuronal populations. Journal of Computational Neuroscience 18(3):297‐309,
2005.
2. B. Naundorf, F. Wolf, and M. Volgushev. Unique features of action potential initiation in
cortical neurons. Nature 440:1060‐3, 2006.
3. M. Timme, T. Geisel, and F. Wolf. Speed of synchronization in complex networks of
neural oscillators: Analytic results based on Random Matrix Theory. Chaos 16:15108, 2006.
4. M. Timme, and F. Wolf. The simplest problem in the collective dynamics of neural
networks: is synchrony stable? Nonlinearity 21:1579--‐1585, 2008.
5. F. Wolf, and E. Bodenschatz. Focus on Heart and Mind. New Journal of Physics
10:015002, 2008.
6. T. Tchumatchenko, M. Volgushev, T. Geisel, and F.Wolf. Correlations and Synchrony in
Threshold Neuron Models. Phys. Rev. Lett. 104:058102, 2010.
7. M. Monteforte, and F. Wolf. Dynamical Entropy Production in Spiking Neuron Networks in
The Balanced State. Phys. Rev. Lett. 105:268104, 2010.
8. S. Junek, E. Kludt, F. Wolf, and D. Schild. Olfactory Coding with Patterns of Response
Latencies. Neuron. 67:872‐884, 2010.
9. T. Tchumatchenko, and F. Wolf. Representation of Dynamical Stimuli in Populations of
Threshold Neurons. PLoS Comput. Biol. 7(10): e1002239, 2011.
10. T. Tchumatchenko, A. Malyshev, F. Wolf, and M. Volgushev. Ultrafast Population
Encoding by Cortical Neurons. J.Neurosci. 31(34): 12171‐12179, 2011.
11. W. Wei, and F. Wolf. Spike Onset Dynamics and Response Speed in Neuronal
Populations. Phys. Rev. Lett. 106:088102, 2011.
Robert Young
University of Hawaii, Honolulu
THE “LILLIE TRANSITION”: MODELING THE ONSET OF SALTATORY
CONDUCTION IN THE DEVELOPMENT AND EVOLUTION OF MYELIN
In 1925, Lillie covered an iron wire model of nerve impulse propagation with segments of
insulating glass tubing. The glass segments and the short interruptions between them
represented the myelin sheath and nodes of myelinated axons, respectively, and induced
saltatory conduction in the model. It was the first demonstration of the likely mode of action
of myelin. A similar transition from continuous to saltatory conduction must have occurred in
the myelination of preexisting unmyelinated axons and the convergent evolution of myelin in
unrelated taxa. We examined this "Lillie transition" by substituting Hodgkin-Huxley squid
axon parameters for the iron wire and using the "extracellular" mechanism for the sheath in
the NEURON simulation environment of Hines, Moore, and Carnevale. To explore the
parametric dependence of this change in conduction mode, the width of the submyelin space
δ, was varied by itself and pairwise with geometric parameters: axon diameter, number of
myelin layers, internode "sheath" length, and node length, over various ranges and the
resulting changes in voltage behavior were recorded.
Axons with large submyelin gaps δ simulated unmyelinated axons and behaved exactly as
predicted by Hodgkin-Huxley. As δ was decreased, action potentials propagated slower as
the resistance of the submyelin space increased. At smaller δ, successive distal nodes
initiated novel action potentials before the arrival of previously existing ones, resulting in
saltatory conduction and a speedup in conduction velocity. The tightness required for a
speedup and the preceding slowdown represent the cost of myelination. The tightening of δ
was costly for axons of small diameter or long internode lengths. Axons with large
diameters, high number of layers, and a node length of about 10 µm benefited most from
decreasing δ.
1,2Yuguo Yu and David A. McCormick1
1Dept. Neurobiology and Kavli Institute for Neuroscience, Yale University School of
Medicine, 333 Cedar Street New Haven, CT 06510 2Center for Computational Systems
Biology, Fudan University, Shanghai People’s Republic of China 200433
DOES HODGKIN-HUXLEY THEORY NEED TO UPGRADE FOR MAMMALIAN
CORTICAL NEURONS?
Since the classic Hodgkin-Huxley (HH) model of action potential generation and propagation
was established using the squid giant axon 60 years ago, it has been challenged several
times. In recent years, in particular, the validity of the HH model for the properties of action
potential generation in mammalian cortical neurons has been questioned. Two well
publicized challenges are: 1) the rapid rising phase and apparent variable threshold of action
potentials in cortical neurons in comparison to the slow rising phase of action potentials
generated by a single compartment HH model [1]; 2) the apparent high energy-efficiency of
action potential generation in mammalian cortical neurons in comparison to those of the
squid giant axon [2,3]. Both of these lines of evidence suggested that the spiking
mechanisms in mammalian neurons and axons may have evolved to be substantially
different from those envisioned by the classical HH theory for squid giant axon. One study in
particular was very bold in asserting that the classical HH theory could not account for the
properties of action potential initiation in mammals [1].
By performing detailed recordings of cortical pyramidal cells and their axons, as well as
exploring HH models of these action potentials, we address both of these areas of
controversy. We demonstrate that the classical HH theories are sufficient to explain the
published observations, and no radical modifications to this theory is needed. Our results
are briefly listed in the following.
Spatial properties of action potential propagation accounts for spike dynamics.
Our findings revealed [4, 5] that models of cortical pyramidal cells based solely on the
principals established by Hodgkin and Huxley can capture the so-called unique features of
action potential generation in cortical neurons observed by Naundorf et al. [1]. We attribute
these so-called unique features simply to recording from a site (e.g. the soma) that is distant
from the site of action potential initiation (e.g. the axon initial segment). This recording from
a distance leads to apparent increased variability in spike threshold owing to differences in
membrane fluctuations, as well as differences in the rate of spike rise, at the two sites. The
recording of action potentials distal from their site of initiation results in the appearance of a
rapid change in dV/dt at spike initiation (the “kink”), owing to the large axial current supplied
by spike generation at the distal site. In other words, it is the propagation of the spike from a
distal site into a local site that results in both a rapid rising phase (“kink”) as well as an
apparent spike threshold variance. We conclude that the observations made by Naundorf et
al. [1] are perfectly in keeping with Hodgkin and Huxley theory and the known physiology of
spike initiation in cortical neurons. There is no need for exotic inter-channel cooperativity
mechanisms to explain their observations.
The temperature dependence of ion channels accounts for energy efficient action potentials in mammalian neurons.
Action potential generation in mammalian, versus invertebrate, axons is remarkably energy
efficient [2, 3]. Here both our computational model based on traditional Hodgkin-Huxley
model and a modified cortical axon model (the kinetics are revised based on our
experimental results) and experimental results [6] demonstrate that temperature is a major
factor which directly modulates the level of energy cost of action potentials. Increases in
temperature result in a large increase in the rate of Na+ channel inactivation, owing to the
Q10 effect. This results in a marked reduction in overlap of the inward Na+, and outward K+
currents for both the classical HH model and our cortical axon model. As a consequence, the
Na+ entry ratio gradually reaches to 1 (the theoretical optimal level, which requires only
minimal Na+ charge for generating an action potential) as temperature rises. Moreover, we
also noticed an exponential increase in firing rate and an exponential decrease in spike
duration with increases in temperature in both our experimental and model studies. The total
energy charge in response to a signal reaches a global minimum between 37-42o C. Our
results suggest that the warm body temperature present in mammals may actually increase
the efficiency of spike generation in the brain.
In summary, although it is clear that there are differences in the precise properties of Na+
and K+ channels underlying spike generation in different neuronal types in different species,
the HH model still provides an accurate framework under which these mechanisms may be
interpreted and understood.
References:
[1] Naundorf, B., Wolf, F. & Volgushev, M. Unique features of action potential initiation in
cortical neurons. Nature 440, 1060-1063. (2006).
[2] Alle H, Roth A, Geiger JR (2009) Energy-efficient action potentials in hippocampal mossy
fibers. Science 325: 1405-1408.
[3] Carter BC, Bean BP (2009) Sodium entry during action potentials of mammalian neurons:
incomplete inactivation and reduced metabolic efficiency in fast-spiking neurons. Neuron 64:
898-909.
[4] McCormick D.A., Shu Y., and Yu Y.G. Variability and rapid rate of rise at action potential
threshold in cortical neurons results from spike initiation in the axon initial segment. Nature.
445: E1-E2, (2007).
[5] Yu Y.G., Shu Y., Duque A., Haider B., and McCormick D.A. Cortical Action Potential
Back-propagation Explains Spike Threshold Variability and Rapid-Onset Kinetics. Journal
of Neuroscience, 28: 7260-7272, (2008).
[6] Yu, Y.G., Hill, A. and McCormick, D.A. Warm body temperature facilitates energy efficient
cortical action potentials. PLoS Comp. Biol., 2012, accepted.
Harold Zakon (University of Texas, Austin), Ashlee Rowe (University of Texas, Austin), Matthew Rowe (Sam Houston State University), Yucheng Xiao (Indiana University School of Medicine), Theodore Cummings (Indiana University School of Medicine).
ADAPTIVE EVOLUTION OF NA CHANNELS IN PAIN RECEPTORS.
Grasshopper mice (Onychomys torridus) from the Sonoran desert prey on the most venomous scorpion in the USA, the Arizona bark scorpion (Centruroides exilicauda). These mice are highly resistant to the deadly effects of scorpion venom and show no sign of pain when stung. Pain-receptive dorsal root ganglion (DRG) neurons express multiple sodium channels, two of which are Nav1.7 and Nav1.8. Nav1.7 currents depolarize DRG pain-sensing neurons sufficiently to activate Nav1.8 currents which generate action potentials (APs). Scorpion toxins inflict pain by activating Nav1.7 causing this channel to be hyperactive and recruiting Nav1.8 to fire action potentials. Scorpion venoms do not activate Nav1.8. Surprisingly, we found that pain resistance in grasshopper mice is mediated by a novel mechanism: the Nav1.8 channel in grasshopper mice is blocked by some C. exilicauda venom peptides. These peptides have the molecular weight of Na+ channel toxins; we are currently determining their amino acid sequences. We show that application of C. exilicauda venom to grasshopper mouse DRGs decreases the number of APs activated by a current pulse. Thus, venom should prevent the propagation of APs to the brain. These data suggest that grasshopper mice essentially convert a potent pain-causing peptide into an analgesic. We demonstrate this in behavioral tests showing that pre-treatment with venom decreases pain responses to formalin, a pain-causing substance. We sequenced Nav1.8 in grasshopper mice and compared it to Nav1.8 of other mammals. We note amino acid substitutions at otherwise highly conserved sites in the channel pore. We are currently investigating whether these substitutions allow scorpion venom peptides to bind to the pore and block the channel.
Delegate
Institution E-mail address
Aihara Kazuyuki Institute of Industrial Science, Japan [email protected]
Archila Santiago Emory University [email protected]
Ashmore Jonathan The Physiological Society & UCL [email protected]
Attwell David University College London [email protected]
Bean Bruce Harvard Medical School [email protected]
Barreto Ernest George Mason University [email protected]
Borkowski Lech Adam Mickiewicz University, Poznan, Poland [email protected]
Boross-Toby Nick The Physiological Society [email protected]
Botcharova Maria University College London [email protected]
Bouchekhima Nacer University of Warwick [email protected]
Bower James University of Texas Health Science Center [email protected]
Braun Hans University of Marburg, Germany [email protected]
Buckley David University College London [email protected]
Cauldron Quentin University of Warwick [email protected]
Constantinescu Andra Princeton University [email protected]
Dervinis Martynas Cardiff University [email protected]
Dua Pinky Pfizer Ltd [email protected]
Faisal Aldo Imperial College London [email protected]
Fernandez Fernando University of Utah [email protected]
Ferster David Northwestern University [email protected]
Finke Christian University of Oldenburg [email protected]
Finn Amber Imperial College London [email protected]
Fliedervish Ilya Ben-Gurion University [email protected]
Forsythe Ian The Physiological Society & University of Leicester [email protected]
Gardner Daniel Weill Cornell Medical College [email protected]
Gardner Esther New York University School of Medicine [email protected]
Giugliano Michele University of Antwerp [email protected]
Glynn Ian University of Cambridge [email protected]
Goodchild Jonathan The Physiological Society [email protected]
Günay Cengiz Emory University [email protected]
Gutnick Michael The Hebrew University of Jerusalem [email protected]
Harris Bill University of Cambridge [email protected]
Hartley Caroline University College London [email protected]
Häusser Michael University College London [email protected]
Hebden Peter University of Warwick [email protected]
Herz Andreas BCCN Munich [email protected]
Hille Bertil University of Washington [email protected]
Hjorth Johannes University of Cambridge [email protected]
Hockings Nick University of Sussex [email protected]
Hodgkin Deborah University of Cambridge [email protected]
Hong Sungho Okinawa Institute of Science and Technology [email protected]
Huang Shiwei Okinawa Institute of Science and Technology [email protected]
Huxley Carol The Physiological Society [email protected]
Jonas Peter IST, Austria [email protected]
Kath William Northwestern University [email protected]
Katz Efrat The Hebrew University of Jerusalem [email protected]
Kitamura Kazuo The University of Tokyo [email protected]
kitano Katsunori Ritsumeikan University, Japan [email protected]
Kurniawan Veldri Cardiff University [email protected]
Landgraf Matthias University of Cambridge [email protected]
Laughlin Simon University of Cambridge [email protected]
Lee David University of Edinburgh [email protected]
Long Lyle Pennsylvania State University [email protected]
Luccioli Stefano Istituto Sistemi Complessi - CNR, Sesto Fiorentino, Italy [email protected]
McDonnell Mark University of South Australia [email protected]
Merrison Robert Plymouth University [email protected]
Migliore Michele CNR, Palermo, Italy [email protected]
Milescu Lorin University of Missouri [email protected]
Neef Andreas MPI Dynamics and Self-Organization, Goettingen, Germany [email protected]
Neishabouri Mohammad Ali Imperial College London [email protected]
Nevo Uri Tel Aviv University, Israel [email protected]
O'Connor Simon Cardiff University [email protected]
Ofer Netanel Bar Ilan university [email protected]
Paulsen Ole University of Cambridge [email protected]
Pearce Tim University of Leicester [email protected]
Rall Wilfrid retired NIH, Bethesda [email protected]
Raman Indira Northwestern University [email protected]
Riichiro Hira National institute of Basic Biology, Okazaki, Japan [email protected]
Robinson Hugh University of Cambridge [email protected]
Roemschied Frederic Bernstein Center for Computational Neuroscience [email protected]
Roth Arnd University College London [email protected]
Rowat Peter University California at San Diego [email protected]
Schleimer Jan-Hendrik Institute for theoretical Biology [email protected]
Schnelle Cornelia The Physiological Society [email protected]
Schreiber Susanne Bernstein Center, Berlin [email protected]
Segev Idan Hebrew University, Jerusalem [email protected]
Sengupta Biswa Indian institute of Science, Bangalore [email protected]
Shepherd Gordon Yale University [email protected]
Singer Joshua University of Maryland [email protected]
Spyer Mike The Physiological Society & NHS London [email protected]
Stemmler Martin Ludwig-Maximilians-Univeristät [email protected]
Sterratt David University of Edinburgh [email protected]
Takahashi Nobuyuki Hokkaido University of Education, Hakodate [email protected]
Takahashi Koko Nobuyuki Takahashi Institute [email protected]
Timofeeva Yulia University of Warwick [email protected]
Timothy Joe University of Manchester [email protected]
Torcini Alessandro CNR, Florence [email protected]
Tuckwell Henry Max Planck Institute, Leipzig [email protected]
van Bogaert Pierre Paul Universiteit Antweren [email protected]
Volgushev Maxim University of Connecticut [email protected]
Wegner Sven University of Manchester [email protected]
Wolf Fred Max Planck Institute, Göttingen [email protected]
Wright Philip The Physiological Society [email protected]
Young Robert University of Hawaii [email protected]
Yu Yuguo Yale University/Fudan University, Shanghai [email protected]
Zakon Harold University of Texas at Austin [email protected]
Zheng Ying University of Sheffield [email protected]
