1. 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

2. 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

3. 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

4. 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*

5. Response to a Step of Current
100 ms
20mV

6. 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

7. 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)

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

9. 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

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

11. Homogeneous Oscillations Regime
Average Voltage
Population

12. 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:

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

14. 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)

15. 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

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

17. 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

18. Equations for the Order Parameters

19. 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

20. 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 !

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

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

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

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

25. 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

26. Qualitative Phase Diagram of the Conductance-Based Ring Model

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