Convergence and stability in networks with spiking neurons

Stan Gielen

Dept. of BiophysicsMagteld Zeitler

Daniele Marinazzo

Hodgkin-Huxley neuron

V mV

0 mV

V mV

0 mV

IC

INa

Membrane voltage equation

-Cm dV/dt = gmax, Nam3h(V-Vna) + gmax, K n4 (V-VK ) + g leak(V-Vleak)

K

V (mV)

mmOpen Closedm

m

mProbability:

State:

(1-m)

Channel Open Probability:

dt

dm m)1( m m m

hhdtdh

hh )1(

Gating kinetics

m.m.m.h=m3h

mm

mm

mm

1

Actionpotential

Simplification of Hodgkin-HuxleyFast variables• membrane potential V• activation rate for Na+

m

Slow variables• activation rate for K+ n• inactivation rate for

Na+ h

-C dV/dt = gNam3h(V-Ena)+gKn4(V-EK)+gL(V-EL) + I

dm/dt = αm(1-m)-βmm

dh/dt = αh(1-h)-βhh

dn/dt = αn(1-n)-βnn

Morris-Lecar model

Phase diagram for the Morris-Lecar model

Phase diagram for the Morris-Lecar model

Linearisation around singular point :

W

V

b

V

W

V

dt

d

1)1( 2

*

*

WWW

VVV

Phase diagram

Phase diagram of the Morris-Lecar model

Overview• What’s the fun about synchronization ?• Neuron models• Phase resetting by external input• Synchronization of two neural oscillators• What happens when multiple oscillators are coupled ?• Feedback between clusters of neurons• Stable propagation of synchronized spiking in neural

networks• Current problems

Neuronal synchronization due to external inputT

ΔTΔ(θ)= ΔT/T

Synaptic input

Neuronal synchronizationT

ΔTΔ(θ)= ΔT/T

Phase shift as a function of the relative phase of the external input.

Phase advance

Hyperpolarizing stimulus

Depolarizing stimulus

Neuronal synchronizationT

ΔTΔ(θ)= ΔT/T

Suppose:

• T = 95 ms

• external trigger: every 76 ms

• Synchronization when ΔT/T=(95-76)/95=0.2

• external trigger at time 0.7x95 ms = 66.5 ms

ExampleT=95 ms

P=76 ms = T(95 ms) - Δ(θ)

For strong excitatory coupling, 1:1 synchronization is not unusual. For weaker coupling we may find other rhythms, like 1:2, 2:3, etc.

Neuronal synchronizationT

ΔTΔ(θ)= ΔT/T

Suppose:

• T = 95 ms

• external trigger: every 76 ms

• Synchronization when ΔT/T=(95-76)/95=0.2

• external trigger at time 0.7x95 ms = 66.5 ms

StableUnstable

Convergence to a fixed-point Θ* requires

Substitution of and expansion near gives

Convergence requires

and constraint gives

TPnnn /)(1

TPnnn /)(1

TP /)( * |||| **

1 nn

n= n* *

nn 1 nn TP

)(

/)(

)( *

1)(1

n

n n

n

1 <1

-1< 1)(<1 and so –2 < )(<0

T

P

Overview• What’s the fun about synchronization ?• Neuron models• Phase resetting by external input• Synchronization of two neural oscillators• What happens when multiple oscillators are coupled ?• Feedback between clusters of neurons• Stable propagation of synchronized spiking in neural

networks• Current problems