Rate models with delays and the dynamics of large networks of spiking neurons David Hansel Laboratoire de Neurophysique et Physiologie CNRS-Université René Descartes, Paris, France With: Alex Roxin and Nicolas Brunel Newton Institute 27/09/05

Investigating Brain Dynamical States Nature of the interactions: Roles of excitation , inhibition, electrical synapses ? How do they cooperate ? Architecture of the network: Footprint, layers .... Dynamics of the interactions: Slow / fast; delays ; depression/facilitation Interplay with intrinsic properties of neurons: e.g. Post-inhibitory rebound; Spike frequency adaptation... ? What shape the spatio-temporal structure of the activity in the brain: e.g. frequency of population oscillations; spatial range of coherent regions. How are brain states of activity on various spatial and temporal scales modulated /controled Relationship between brain dynamics and Functions and Dysfunctions of the CNS

The Ring Model: Architecture NE excitatory neurons; NI inhibitory neurons on a 1-D network with periodic boundary conditions (ring). Each neuron is caracterized by its position on the ring, x: - < x < . The synaptic weight / connection probability from neuron (y,to neuron (x, is a function of |x-y| and of ,J (|x-y|). Note:a particular case is all to all connectivity: J(x-y)=J0 The neurons receive an external input I (x,t). + Dynamics of the neurons and of the synapses Model of local circuit in cortex

Conductance-Based Ring Model 1- Dynamics: neurons described by Hodgkin-Huxley type model. C dV/dt = - IL – currents of all active channels - Isyn + Istim + equations for gating variables active channels e.g Na, K ... Isyn : synaptic current from other neurons in the network Istim : external noisy stimulus 2- Synaptic current: Isyn = -g s(t) ( V – Vsyn) After a presynaptic spike at t* : s(t) ---> s(t)+ exp[(-t +t*)/tsyn ] for t > t*

Response to a Step of Current 100 ms 20mV

4 Replace the conductance-based model by a rate model/neural field Approaches 1 Numerical simulations -2 Reduction to phase models: Assume weak coupling, weak noise, weak heterogeneities -3 Replace conductance-based by integrate-and-fire dynamics and use Fokker-Planck approach to study the stability of the asynchronous state Note: If F(V)= -g V (leaky integrate-and-fire) noise and sparse connectivity can be included. 4 Replace the conductance-based model by a rate model/neural field

The Ring Model: Rate Dynamics The dynamical state of a neuron x in the population  = E,I is caracterized by an activity variable m(x,t). (x) is the non-linear input-output neuronal transfer function;  is the time constant of the rate dynamics of population  JEE(x)= JIE (x) JII (x)= JEI (x) E= I =  IE= II mE (x,t) = mI(x,t) = m(x,t) Effective coupling: J(x)= JEE(x) - JEI(x) = JIE(x) - JII(x)

J(x)= J0 + J1 cos (x) F(h) threshold linear i.e F(h) = h if h >0 and zero otherwise

The Phase Diagram of the Reduced Ring Model for an homogeneous external input IE= II independent of x J(x)= J0 + J1 cos (x) m(x,t)=m0

Reduced Ring Conductance-Based Model Asynchronous State Stationary Bump 200 msec 200 msec

Homogeneous Oscillations Regime Average Voltage Population

Reduced One Population Ring Model with Delays Minimal rate model with delays: J(x)= J0 + J1 cos (x) F(h) threshold linear Axonal propagation Dendritic processing Synaptic dynamics Spikes dynamics Sources of delays in neuronal systems:

The Phase Diagram of the Rate Model: D=0.1 t «epileptic » 

Instabilities of the Stationary Uniform State The stationary uniform state, m(x,t) = m0 , is a trivial solution of the dynamics. The dispersion relation for the stability of this state is: l= -1 + Jn exp(- l D) With: Hence for J(y)=J0 +J1 cos(x) there are 4 types of instabilities: -Rate instability (w=0, n=0) for J0=1 -Turing instability (w=0, n=1) for J1=2 Hopf instability (w > 0, n=0) for J0 cos(w D)=1 with w = -tan (w D) -Turing-Hopf instability (w > 0, n=1) for J1 cos(w D)=1 with w=-tan(w D)

The Stationary Bump and its Instabilities Like for D=0: The stationary uniform states looses stability via a Turing instability when J1 crosses 2 from below. The resulting state is a stationary bump (SB). Self-consistent equations for the bump caracteristics and stability can be computed analytically. Strong local excitation  rate instability  Neurons go to saturation Homogeneous oscillations -Strong inhibition  oscillatory instability Localized synchronous activity

Bump of Synchronous Oscillatory Activity Rate Model Conductance-Based Model 200 ms

The Oscillatory Uniform State If J0 is sufficiently negative the stationary uniform state undergoes a Hopf bifurcation with a spatially uniform unstable mode. D/t <<1 : bifurcation at J0 ~ -p/2 t/D; frequency of unstable mode is : f ~ ¼ t/D. The amplitude of the instability grows until the total input to the neurons, I tot, becomes subthreshold. Then it decays until I tot =0+… The activity of the network remains uniform but it is now oscillatory. D/t = 0.1 m(t) t/t

Equations for the Order Parameters

The Oscillatory Uniform State and its Stability The homogeneous limit cycle can be explicitely constructed: Step 1: 0<t < T1 : Itot< 0 and m(t) ~ exp(-t/t); T1 defined by Itot(T1 )=0+ Step 2: T1 < t < T1 +D: m(t) satisfies: t dm/dt = -m + Itot(t-D)  m(t) = A exp(-t/t) + particular solution driven by the value of m in the previous epoch 0<t< T1 Repete Step 2 for as many epochs are required to cover the full period of the limit cycle, T; T is determined by the self-consistent condition m(T)=m(0) and Itot(T)=0. Stability can be computed analytically: Step 1: Linearize the order parameter dynamics Step 2: Integrate in each epochs of the limit cycle using the fact that F’(x)=Heaviside(x) to determine the Floquet exponents of the limit cycle

The Oscillatory Uniform State: Results of the Stability Analysis There are in general two Floquet exponents; e.g. assuming T<2 D: With R=T-T1< D. This can be extended for arbitrary T. b0=1: corresponds to the time translation invariance on the homogeneous limit cycle b1 corresponds to the spatially heterogeneous mode cos(x) Stability iff | b1 [< 1. b1=-1  period doubling instability with spatial modulation b1 =1  phase instability of the spatially heterogeneous mode. This occurs in particular on the line: J1=2J0: NOTE: Numerical simulations show that these instabilities are subcritical !

The Standing Waves For sufficiently strong modulated inhibition standing waves are found Rate Model Conductance-Based Model 25 ms

Interaction of the Standing Waves with a Non Homogeneous Input Regime 1 Regime 2

Chaotic State in the Rate Model) Order parameters are aperiodic Local activity: auto and cross correlations

Chaotic State in the Conductance-Based Model t(msec)

Travelling Waves Assuming m(x,t)=m(x-vt) once can derive self-consistent equations for the profile of the wave and the velocity v and for the stability of the pattern. In the conductance based model we were unable to find stable waves

Qualitative Phase Diagram of the Conductance-Based Ring Model

Bistability A 30 msec inhibitory pulse applied to 500 neurons switches The network state from homogeneous oscillations to a standing wave