Convergence and stability in networks with spiking neurons Stan Gielen Dept. of Biophysics Magteld Zeitler Daniele Marinazzo

Hodgkin-Huxley neuron

V mV 0 mV V mV 0 mV ICIC I Na Membrane voltage equation -C m dV/dt = g max, Na m 3 h(V-V na ) + g max, K n 4 (V-V K ) + g leak (V-V leak ) K

V (mV) mm mm Open Closed mm mm m Probability: State: (1-m) Channel Open Probability: mm mm Gating kinetics m.m.m.h=m 3 h

Actionpotential

Simplification of Hodgkin-Huxley Fast variables membrane potential V activation rate for Na + m Slow variables activation rate for K + n inactivation rate for Na + h -C dV/dt = g Na m 3 h(V-E na )+g K n 4 (V-E K )+g L (V-E L ) + I dm/dt = α m (1-m)-β m m dh/dt = α h (1-h)-β h h dn/dt = α n (1-n)-β n n Morris-Lecar model

Phase diagram for the Morris-Lecar model

Linearisation around singular point :

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 input T ΔT Δ(θ)= ΔT/T Synaptic input

Neuronal synchronization T ΔT Δ(θ)= ΔT/T Phase shift as a function of the relative phase of the external input. Phase advance Hyperpolarizing stimulus Depolarizing stimulus

Neuronal synchronization T ΔT Δ(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

Example T=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 synchronization T ΔT Δ(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 Stable Unstable

Convergence to a fixed-point Θ * requires Substitution of and expansion near gives Convergence requires and constraint gives 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