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Multi-scale modelling of neural circuits BRANDY School Val di Sole

Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

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Page 1: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Multi-scale modelling of neural circuits

BRANDY SchoolVal di Sole

Page 2: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Also known as

Comp Neuro 101

Page 3: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Why do we need models?

§ Common misunderstanding: Modelling is not a form of hypothesis testing: “garbage in, garbage out”

§ Forces us to make assumptions explicit. § Enables many “virtual" experiments to be done, can

pinpoint the one that is most crucial.§ Can lead to unexpected predictions.§ Often much quicker/easier to try out ideas, e.g. blocking

a connection or channel type

Page 4: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

What makes a good model?§ However, it's easy to make a bad alifey or

physicsy model§ Good to have close contact with

neuroscientists§ Model should not only replicate existing

data but must also make new predictions about the biological system

Page 5: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Scales in the nervous systemLevels of modelling (1)

Page 6: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Levels (scales) of modelling§ Many different types of models: There is a

continuum from very realistic to very abstract. All models must make simplifications to be useful.

§ For instance, a model of a single neuron:§ Binary threshold unit, continuous unit § Rate model, Integrate-and-fire model, Spiking neuron§ A few compartments, many compartments§ Individual channels, detailed model of channel dynamics§ …

Page 7: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Which model to use?§ Depends on purpose of the model! Different types

are appropriate for different sorts of questions.§ Two mainstream approaches: Top-down vs bottom

up:§ Top-down: based on rational theoretical hypothesis to infer

the mechanism of brain functions, i.e., start with an idea about abstract task / problem, figure out a good way to solve it and see if that's what the nervous system does.

§ Bottom-up: based on experimental data, i.e., look closely at the nervous system, try and figure out what it’s doing, derive the task/problem from there.

Page 8: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Modelling neuronsNeurons have a soma (cell body), dendrites (input) and axon (output).

We will abstract this to ”single compartment” or “point” neurons

dendrites

soma

axon

Page 9: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Ion channelsIon channels in the membrane lower the effective membraneresistance by a factor of 10,000 (depending on density, type, etc.)Ion pumps maintain the

differences in concentrations inside and outside by expending energy.

Page 10: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

The membrane potential

Difference in concentration: ions diffuse through channels (when open)

Exterior potential 0 by convention

Excess of positive charge outside -> resting membrane potential is negative.

Page 11: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Conductance based model

If Cm is constant:

Page 12: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Adding ion channels

synap

ticca

pacit

ivelea

kage

Ion ch

anne

l Ion

chan

nel

Ionic current

maximal conductance

activation reversal potential

Page 13: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Persistent and transient currents

Delayed rectifier Sodium current

Page 14: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Hodgkin-Huxley model

Non-inactivating delayed rectifier current

Page 15: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Hodgkin-Huxley modelSodium current

“activation”

“inactivation”

Page 16: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Hodgkin-Huxley modelIs often rewritten in terms of

⇒ dn

dt=

1

τn(n∞ − n)

Page 17: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Hodgkin-Huxley model

Page 18: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Action potential

+30 mV

0 mV

-55 mV

-70 mV

-90 mV input

Naactivation

Nainactivation

Kdactivation

Nadeactivationde-inactivation

Kddeactivationm → 1

m → 0h → 1

h → 0 n → 1

n → 0

depo

laris

atio

n

repo

laris

atio

n

Page 19: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

ActionPotential

Page 20: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Hodgkin-Huxley model§ The original model is “numerically not very

nice”§ Is a type II neuron (Hopf bifurcation)§ Today, not used much§ One of the most-used models is the HH-like

model by Traub & Miles (1991), which has type I behaviour (saddle-node bifurcation)

Page 21: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Traub-Miles model§ In practice, it works very well§ Surprisingly, it has numerical instabilities:

Unclear at V=-50. § L’Hôpital shows that it’s well-defined and

continuous

Page 22: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Modelling networks§ Action potential travel down axons

(mostly not modeled explicitly)§ Axons make contacts with target

neuron’s dendrites at Synapses§ Synapses can – depending on

transmitter – be excitatory (depolarize target neuron) or inhibitory (hyper-polarize)

Page 23: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

SynapsesSynaptic transmission is by chemical transmitters for most synapsesDepolarization of the pre-synaptic terminal leads to transmitter release across the cleftPost-synaptic channels are opened by the transmitter

Page 24: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Modeling synapses

Fraction s of released transmitter:

Synaptic current into post-synaptic neuron:Isyn = gsyns(Vrev − Vpost)

ds

dt= −βs + α(1 − s)1[t,t+tr]

Page 25: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

How to use HH models§ Used for numerical simulations:

§ Each neuron:

§ Each pre-synapse:

§ Each post-synapse:

dVj

dt= . . .

dsjdt

= . . .

Ii,syn =!

j

gijsj(Vrev − Vi)

Page 26: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

How to simulate …§ Euler Algorithm

§ Good if equations not too stiff, small time steps

§ Otherwise: Runge-Kutta algorithms, implicit algorithms, …

Vi(t + ∆t) = V (t) +dVi

dt∆t + O(∆t2)

Page 27: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Simplified models§ One can work with V as the main

variable (and many people/studies do)§ But for larger scale/ different analysis

one can use simplified models

Page 28: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

A zoo of models …§ Morris-Lecar§ Fitzhugh-Nagumo§ (leaky) integrate-and-fire

(IF/ LIF), exponential, adExp, GIF

§ McCulloch-Pitts§ Hindmarsh-Rose§ Izhikevich, Rulkov§ Wilson-Cowan§ Kuramoto

Reduced HH

discrete time/ probabilistic“abstract spiking”

rate modelsphase oscillators

Page 29: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Integrate-and-Fire

§ Simple membrane equation – passive only

§ Explicit spiking mechanism:if : spike fired, reset:

V = −gL(VL − V )

V > Vth

V → Vreset

Page 30: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Integrate-and-Fire§ Faster to simulate§ Allows some analytical work:

§ Event - based simulation§ General analysis with theory of stochastic

processes

§ Suitable for neuromorphic devices

Page 31: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Different views of the same activity§ For synaptic transmission only spikes matter

Trace of a B1 neuron in the pond snail

*

Detect spikes

Convolve with a kernel

Equivalent “spike density function” – a measure of spike rate

Page 32: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Quantitative rate reduction§ We start from a full, conductance-based

model:

§ Synapses are modelled by

Buckley & Nowotny, PRL 106, 238109 (2011)

CVi = −INa − IK − IL − IM − Ii,ext − Ii,syn

Ii,syn =!

j

gijsj(Vi − Vrev)

si = −βsi + α!

k

1[tk,tk+tr ]

Page 33: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Step 1: f-I curveIt is common to characterize neurons by their so-called F-I curve:• For type 1 (saddle-

node bifurcation) neurons:

• For type 1 with adaptation: linear

no adaptation

with adaptationF (I) ∼

!I − I0

F (I) = [mI + C]+

Page 34: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Step 2: Effective synapse activation

si = −βsi + αtrF

Synapse activation driven by a constant spike train with frequency F:

Page 35: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Step 3: Putting it all together

Insert the F-I curve into the s equation:

s = −βs+ αmtr[−Gs+ θ + I]+

s = −βs+ γ[−Gs+ θ + I]+

F = m[−Gs+ θ + I]+

Page 36: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Exampless F

Page 37: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

So what?§ The rate equations can be further

reduced to mean field models§ These can be analyzed analytically§ E.g. investigate global stability /

“dynamical systems notion” of criticality – see Buckley & Nowotny, PRL 106, 238109 (2011)

Page 38: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Example result§ Maximal dynamic range close to a bifurcation

(“minimally stable global fixed point”)

Page 39: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Software§ There are many packages for neuronal

network simulation:§ Neuron, Genesis§ Brian/ Brian 2§ NEST§ GeNN (GPU enhanced neuronal networks),

Brian2GeNN, SpineML2GeNN

Page 40: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Origins of GeNN§ In 2010 I tried to implement a network on CUDA from scratch§ Achieved a 24x speed up over CPU§ It took a month to implement an existing model (after learning how to

use CUDA)§ The program was optimised for a particular GPU§ It was designed for one size of the simulation

Page 41: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

How GeNN works

Page 42: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Latest version outperforms HPC

Page 43: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Simulation performance

~ 80,000 neurons300M static synapses

Page 44: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Network setup time

Page 45: Multi-scale modelling of neural circuits Tutorial.pdf · Levels (scales) of modelling § Many different types of models: There is a continuum from very realistic to very abstract

Next time …§ A bit about insect olfaction and a odour

object recognition§ Some introduction to our work in the

Brains on Board project regarding insect navigation and autonomous robots