Computational Neuroscience - rmki.kfki.hubanmi/sote/singlecell2.pdfLengyel Máté: Egysejt modellek II / Modellek az idegrendszer-kutatásban 2018 Neuroinformatika, Zalányi László

Lengyel Máté: Egysejt modellek II / Modellek az idegrendszer-kutatásban2018 Neuroinformatika, Zalányi László

Computational Neuroscience

Structure – Dynamics – Implementation – Algorithm – Computation - Function

Lengyel Máté: Egysejt modellek II / Modellek az idegrendszer-kutatásban2018 Neuroinformatika, Zalányi László 2

Simplified neuron modelsReminder: the HH-model

-100 50V [mV]

m∞

n∞

h∞

m

n

h

0

9

[mse

c]

0

1

[1]

-100 50V [mV]

Cm

dVdt

=gNa ENa−V t m3 t h t gK EK−V t n4 t

g leak Eleak−V t I external t

dmdt

=m∞ V t −m t

m V t

dhdt

=h∞ V t −h t

h V t

dndt

=n∞ V t −n t

n V t

One spike

0 50t [msec]

40-80

V [mV]

10

x [1]n

hm


The Hodgkin-Huxley model / 4

3

HH model in work:

membranepotential

gating variables

channelconductances

channel currents

0 50t [msec]

40-80

V [mV]

10

x [1]40

0g [S/cm

2]800

-800I [nA/cm

2]

n

h

K

Na

m

Na

Na

K

leak.



4

Sir John Carew Eccles, Alan Lloyd Hodgkin, Andrew Fielding Huxley: Awarded by Nobel-prize in medicine, 1963.



5


Types of attractors

A: Fixed point

B: Limit cycle

C: Quasi periodic:requires irrational ratio for at leasttwo osscillations.

D: Chaotic: requires at least 3 dimensions.


40-80 V [mV]

10

m [1]

10

h [1]1

0n [1]

m∞

n∞

h∞

0 1h [1]

01

n [1

]

The dimension reduction of the HH-modelOne spike in the state space (three projections)

FAST ACTIVATING VARIABLES (V)

7

SLOW, INACTIVATING VARIABLES(W)

Cm

dVdt

=gNa ENa−V t m∞

3 t 1−W t

gK EK−V t W t s

4

gsziv Eleak−V t I external t

dWdt

=W∞ V t −W t

W V t


55-85 V [mV]

01

W [1

]

V nullclinedVdt

=0

W nullclinedWdt

=0

fixed point

Phase Plane Analysis: the FitzHugh- (Nagumo-, Rinzel-) model / 1The phase plane with the direction field

8


20 mV5 msec

55-85 V [mV]

01

W [1

]Excitability: firing is a threshold phenomenon (all-or-none)

Phase Plane Analysis: the FitzHugh- (Nagumo-, Rinzel-) model / 2

9


20 mV5 msec

55-85 V [mV]

01

W [1

]Periodic firing to constant depolarization


10


20 mV5 msec

Post Inhibitory Rebound: firing to transient hyperpolarization


11

55-85 V [mV]

01

W [1

]


01

W [1

]

55-85 V [mV]

01

W [1

]0

1W

[1]

gK=18 mS/cm2

gK=15 mS/cm2

gK=8 mS/cm2

200 gK [mS/cm2]

-85

60V

[mV]

d gK

dt=Z V V−V rest−

gK−gKrest

gK

Phase Plane Analysis: Bursting / 1

12


Phase Plane Analysis: Bursting / 2

Endogeneous Burster

gKrest

=8mS /cm2 gKrest

=12mS /cm 2

Conditional Burster


Types of bifurcations

Class 1 ~ saddle-node on invariant circleClass 2 ~ saddle-node or Andronov-Hopf


Types of bifurcations 2.


V

Wt

d

dt= I t h t , t '

V

W

t

d

dt= I t , t h t , t '

The phase-model


The rate-modelType I Type 2 (HH)

r t =T ∑i=1

n

wi⋅x i t − T

r

tx

i

xn

wn

wi

w1

r


y t =H ∑i=1

n

w i⋅xi t −xi

xn

wn

wi

w1

y

H1

The McCulloch-Pitts model

x1


Integrate & Fire neuron

Self-flushingLavatory tank

Cm I

ext

V Cm

dV t dt

= I ext t

ha V(t)>Vthr

spike V(t):=Vret

V

t

Vthr

Vret

Leaky Integrator

Iext

V

Cm

Rm= 1/g

leak

Eleak

Cm

dV t dt

=E leak−V t

Rm

I ext t V

t

Vthr

Vret

19


Resonate & Fire neuron

Cm

dZ t dt

=biZI ext t

if V(t)>Vthr

spike V(t):=Vret

20

Re(Z) is a potential-, Im(Z) is a current-like variable

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Computational Neuroscience - rmki.kfki.hubanmi/sote/singlecell2.pdfLengyel Máté: Egysejt modellek II / Modellek az idegrendszer-kutatásban 2018 Neuroinformatika, Zalányi László