JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models

SS2016 Modern Neural Computation

Lecture 1: Single Neurons

Hirokazu TanakaSchool of Information Science

Japan Institute of Science and Technology

Neuron as a computational unit of the brain.

In this lecture we will learn:• Basic anatomy and physiology of neuron

- morphology- membrane properties

• Phenomenological models with subthreshold dynamics- Integrate-and-fire model, Quadratic-and-fire model, Resonate-and-fire model

• Biophysical models with spiking mechanism- Ion channels, master equations- Hodgkin-Huxley model

• Phase plots and bifurcation analysis- Saddle-node bifurcation, Andronov-Hopf bifurcation- FitzHugh-Nagumo model, Hindmarsh-Rose model

• Modern single-neuron models- Izhikevich model, Adaptive-exponential model

Neurons composed of dendrites, soma and axon.

Figure 3.1, Fundamental Neuroscience, 3rd Edition

Morphology: Neurons take various shapes.

Figure 2.1, Fundamental of Computational Neuroscience

Cortical neurons receive cortico-cortical and thalamo-cortical inputs.

Figure 3.2, Fundamental Neuroscience, 3rd Edition

Pyramidal cell in layer II/III

Apical dendrites

Basal dendrites

Lipid-bilayer membrane insulates a neuron

Ruye Wang, http://fourier.eng.hmc.edu/e180/lectures/signal1/node2.html

Physiology: Neurons are electrically excitable.

Figure 2.2, Neuroscience 3rd Edition

Physiology: Neurons take various spiking patterns.

Izhikevich (2004) IEEE Neural Networks

Leaky Integrate-and-fire model (LIF)

( ) ( )( ) ( )m L

dv tv t E RI t

dtτ = − − +

( ) ( )f

f

j

j jj t

I t w t tα= −∑∑

( )fthv t V=

( )freset .v t V←

Leaky integration Fire (spike)If the potential reaches the threshold voltage,

then, add a spike and reset the potential to the reset voltage.


Lapicque (1907)For English translation, see:

Brunel & van Rossum (2007)

Analytical solution of LIF model with constant current.

( ) ( )m

dv tv t RI

dtτ = − +

( ) ( ) m m0 1 t t

tv t v e RI e RIτ τ

− −

→∞

= + − →


( ) ( )( ) const.

Lv t v t E

I t I

← −

= =

subtracting the equilibrium potential.

considering a time-invariant current.

f-I curve of LIF model.


( )L r

ref mL th

1

lnf I

RI E VRI E V

τ τ=

+ −+ + −

Quadratic-and-fire model

( ) ( )2dv tI v t

dt= +

( ) ( )thresholdif , then resetv t v v t v≥ ←

( ) ( , )dv t

F v Idt

=

In general, the dynamics for membrane potential has a general form:

Quadratic-and-fire (QIF) model: F is quadratic in terms of v and linear in terms of I.

For LIF model, F is linear in terms of both v and I.

( , )F v I v I= − +

Quadratic-and-fire model

Figure 3.35, Dynamical Systems in Neuroscience

Resonate-and-fire model: oscillatory sub-threshold dynamics.

( ) ( )( ) ( )

( ) ( )( ) ( )

leak leak

1/2

dv tC I g v t E w t

dtv t vdw t

w tdt k

= − − −

−= −

For some neurons, the sub-threshold dynamics exhibits an oscillatory behavior:

Resonate-and-fire model: two-dimensional model of membrane potential (v) and the recovery variable (w).

Whole-cell recording of an olivary neuron

Hutcheon & Yarom (2000) Trends Neurosci

Resonate-and-fire model.

Izhikevich (2001) Neural Networks

spike

spikeno spike

no spike

Ion channels: Nernst equation.


[ ][ ]

oution in out

in

ionln

ionRTE E EzF

≡ − =

EoutEin Nernst equation

[ ][ ]

( )out

out in

in

out

in

ionion

zF E zFRT E ERT

zF ERT

e ee

−− −

−

= =

in[ ] 140mMK + = out[ ] 3mMK + =[ ][ ]

out

in

3ln 61.5ln 102mV140K

KRTEF K

= = = −

Potassium ion

Ion channels: Goldman-Hodgkin-Katz equation.


K Na Clout out inm out in

K Na Clin in out

K Na Clln

K Na Cl

p p pRTV V VF p p p

+ + −

+ + −

+ + = − = + +

Goldman-Hodgkin-Katz equation

K Na Cl: : 1.00 : 0.04 : 0.45p p p =

Permeability

For T=293K (20°C), the equilibrium potential is

m out in 62mVV V V= − = −

Ion-channel kinetics: voltage-dependent ion channels

: activation variablen

( )( ) ( )1n ndn V n V ndt

α β= − −Inactive Active

( )n Vα

( )n Vβ

( )activeP n=( )inactive 1P n= −

Master equation

( ) ( )ndnV n V ndt

τ ∞= −

( ) ( ) ( )1

mn n

VV V

τα β

=+

( ) ( )( ) ( )

n

n n

Vn V

V Vα

α β∞ =+

time constant

asymptotic value

Gating equation

Hodgkin-Huxley model: potential and gating dynamics.

( ) ( ) ( )4 3K K Na Na L L

dVC g n E V g m h E V g E V Idt

= − − − − − − +

( ) ( )ndnV n V ndt

τ ∞= −

( ) ( )mdmV m V mdt

τ ∞= −

( ) ( )hdhV h V hdt

τ ∞= −

Membrane-potential dynamics

Gating equations

Figure 5.10, Theoretical Neuroscience

Hodgkin-Huxley model: activation and inactivation variables.

( ) ( ) ( )4 3K K Na Na L L

dVC g n E V g m h E V g E V Idt

= − − − − − − +

( ) ( )ndnV n V ndt

τ ∞= −

( ) ( )mdmV m V mdt

τ ∞= −

( ) ( )hdhV h V hdt

τ ∞= −

Membrane-potential dynamics

Gating equations m: Na+ activation variableh: Na+ inactivation variablen: K+ activation variable


m=0h=1

m=1h=1

m=1h=0

Hodgkin-Huxley model reproduces spike waveform.

Figure 5.10, Theoretical Neuroscience Figure 4.3, Neuroscience 3rd Edition

Hodgkin-Huxley model reproduces spike waveform.


Phase-plane plot: one-dimensional case

( ),dV F V Idt

=

*( , ) 0 fixed pointF V I = →( )( )

*

*

, 0 stable (attractive) fixed point

, 0 unstable (repulsive) fixed point

F V I

F V I

′ < →

′ > →


Phase-plane plot: schematic method for capturing qualitative behaviors of differential equations without solving.


Bifurcation: Saddle-node bifurcation

( )dV F V Idt

= +


Phase-plane plot: two-dimensional case

( )( )

,,

V F V ww G V w = =

Phase-plane plot: vector field (dV/dt, dw/dt) on the two dimensional plane.


1, 0x y= = 0, 1x y= =

, x x y y= − = − , x y y x= − = −

Phase-plane plot: Nullclines

( )( )

,,

V F V ww G V w = =

Nullclines: the curves of F(V,w)=0 and G(V,w)=0.


Phase-plane plot: linear stability analysis

( )( )

,,

V F V ww G V w = =


Dynamical Systems with Applications using MATLAB

Stable node Unstable node Saddle point

Unstable focus Stable focus Center

Phase-plane plot: Separatrix

( )( )

,,

V F V ww G V w = =



Separatrix: the boundary separating two modes of behaviour in a differential equation.

Bifurcation: Saddle-node bifurcation

Figure 4.26, 28, 30, Dynamical Systems in Neuroscience

Bifurcation: (supercritical) Andronov-Hopf bifurcation

Figure 4.26, 28, 30, Dynamical Systems in Neuroscience

Class I and II neurons and bifurcation type


Class I: Continuous F-I curve, Saddle-node bifurcationClass II: Discontinuous F-I curve, Andronov-Hopf bifurcation

Two-dim. model: FitzHugh-Nagumo model

( )

3

30.08 0.8 0.7

vv v w I

w v w

= − − +

= − +

FitzHugh (1961) Biophysical J; “FitzHugh-Nagumo Model” (2015) Encyclopedia of Comp Neuro

stable unstable

*I I< *I I<

Two-dim. model: FitzHugh-Nagumo model

( )3

, 0.08 0.8 0.73vv v w I w v w= − − + = − +

FitzHugh (1961) Biophysical J; “FitzHugh-Nagumo Model” (2015) Encyclopedia of Comp Neuro

All-or-nothing response Post-inhibitory spike

Two-dim. model: Hindmarsh-Rose model

( )( )

v f v u I

u g v u

= − +

= −

( ) ( )3 2 2,f v av bv g v c dv= − + = − +

Hindmarsh & Rose (1982) Nature; (1984) Proc R Soc Lond B

Izhikevich model: quadratic and linear nullclines.

thresholdif 1, then and .v v v c u u d≥ = ← ← +

Quadratic v-nullcline and linear u-nullcline can describe both saddle-node and Andronov-Hopf bifurcations.


( )

2v v u Iu a bv u= − +

= −

Izhikevich model reproduces various spiking patterns.

( )( )

20.04 5 140v v v u I t

u a bv u

= + + − +

= −

thresholdif 30, then and .v v v c u u d≥ = ← ← +

Izhikevich (2003) IEEE Trans Neural Networks

Izhikevich model reproduces various spiking patterns.

Izhikevich (2003) IEEE Trans Neural Networks

Adaptive-exponential model

( ) ( )

( )

T

T

V V

m L T L T

w L

dVC g V V g e w I tdtdw a V E wdt

τ

−∆= − − + ∆ − +

= − −

Brette & Gerstner (2005) J Neurophysiol

The adaptive-exponential model are popular to neurophysiologists because …

- It has a form similar to conventional two-dimensional models

- Its parameters are physiologically interpretable.

What we left out: Neuron morphology (shape) does influence physiology (function)!

Mainen & Sejnowski (199) Nature

250 μm

100 ms

25 mV

What we left out: Neuron morphology (shape) does influence physiology (function)!

Branco et al. (2010) Science

Cable equation describes spike propagation.

“Cable Equation” (2015) Encyclopedia of Computational Neuroscience

Rall model reduces to equivalent cylinder model.

“Equivalent Cylinder Model” (2015) Encyclopedia of Computational Neuroscience

With a set of assumptions about the morphological and electrical properties of dendrites, the complex branching structure of a dendritic tree can be reduced to a simple conductive cylinder.

2 23 3

1 22

30

GR d d

d

+=

If GR=1, then the cylinders 1 and 2 can be reduced to a single cylinder.

Conclusions

- Neurons have a wide range of morphology (shapes) and physiology (functions).

- Many fundamental properties of subthreshold dynamics and spiking patterns can be captured by low-dimensional models.

- Models vary in their complexities: from a simple LIF model (just integrating and thresholding) to biophysically detailed Hodgkin-Huxley model.

- Phase-plane and bifurcation analyses are the powerful tool for understanding qualitative behaviors of a dynamical system without an explicit solution.

Exercise

1. Read the following paper and derive a low-dimensional neuron model from a detailed HH-type model by linearizing around the resting potential.

Richardson et al. (2003) “From subthreshold to firing-rate resonance,” J Neurophysiol 89, 2538-2554.

2. Examine a qualitative behavior of the Izhikevich model by plotting a phase portrait:

a=0.02, b=0.2, c=-65, d=6, I=14 (constant).Then confirm your phase-plane analysis with the matlab code provided from Izhikevich’s site:http://www.izhikevich.org/publications/whichmod.htm#izhikevich

Exercise

1. Simulate an integrate-and-fire model using the Euler method and evaluate how accurate the solution is. The Euler method is the simplest numerical integration method.

Brette, R., Rudolph, M., Carnevale, T., Hines, M., Beeman, D., Bower, J. M., ... & Zirpe, M. (2007). Simulation of networks of spiking neurons: a review of tools and strategies. Journal of computational neuroscience, 23(3), 349-398.

2. Simulate the Izhikevich model using standard parameters. Then plot the phase portraits in two dimensions.

( ) ( ) ( )( )m

v t RIv t v t

tt

τ− +

+∆

=∆+

(0.02, 0.2, -65, 8) (0.02, 0.2, -55, 4) (0.02, 0.2, -50, 2) (0.1, 0.2, -65, 2)

(0.02, 0.25, -65, 0.05) (0.02, 0.2, -65, 0.05) (0.1, 0.26, -65, 8) (0.02, 0.25, -65, 2)

% params for RS neuron:a = 0.02; b = 0.20; c = -65.; d = 8;

dt = 1/1000; % integration time step (s)T = 50.; % total simulation time (s)t0 = 0:dt:T; % time steps

%%v = zeros(length(t0),1); % voltage variableu = zeros(length(t0),1); % recovery variableI = zeros(length(t0),1); % input currentI(t0>1000/1000) = 150; spikes = zeros(length(t0),1); % spike timings

v(1) = -80; % initial voltageu(1) = 0; % initial recovery

for n=1:length(t0)-1v(n+1) = v(n) + dt*(0.04*v(n)^2+5*v(n)+140-u(n)+I(n));

% v(n+1) = v(n) + dt/2*(0.04*v(n)^2+5*v(n)+140-u(n)+I(n));% v(n+1) = v(n+1) + dt/2*(0.04*v(n+1)^2+5*v(n+1)+140-u(n)+I(n));

u(n+1) = u(n) + dt*a*(b*v(n)-u(n));

if v(n+1)>30v(n+1) = c;u(n+1) = u(n+1) + d;spikes(n+1) = 1;

end

end

figure(1); clf; subplot(211); plot(t0, v, 'k');subplot(212); plot(t0, u, 'k');

References

• Squire et al. (2008) “Fundamental Neuroscience,” Academic Press.

• Purves et al. (2004) “Neuroscience,” Sinauer Associates.

• Trappenberg (2010) “Fundamentals of Computational Neuroscience,” Oxford University Press, Chapters 2 & 3.

• Dayan & Abbott (2000) “Theoretical Neuroscience,” MIT Press, Chapter 5.

• Izhikevich (2007) “Dynamical Systems in Neuroscience,” MIT Press, Chapters 3 & 4.

Education

JAISTサマースクール2016「脳を知るための理論」講義01 Single neuron models