The Ring Model of Cortical Dynamics: An overview David Hansel Laboratoire de Neurophysique et Physiologie CNRS-Université René Descartes, Paris, France Leiden 22/05/08 Uninet

Investigating Brain Dynamical States Nature of the interactions: Roles of excitation, inhibition, electrical synapses ? How do they cooperate ? 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 Architecture of the network: Footprint, layers....

Response to Elongated Visual Stimuli of Neurons in Primary Visual Cortex (V1) is Orientation-Selective

Anderson et al Science 2000   Orientation Tuning Curves of V1 Neurons Tuning width is contrast Invariant high contrast medium low Drifting Grating grating contrast Preferred Orientation of the Neuron

Functional Organisation of V1 Connectivity is selective to preferred orientations of pre and Postsynaptic neurons  J(|  –  ’|) Neurons with similar PO tend to interact more

Orientation Columns and Hypercolumns in V1

prestimulus - 1 sec. time Delay Activity During ODR Task

prestimulus - 1 sec. cue sec. time Delay Activity During ODR Task

prestimulus - 1 sec. cue sec. delay - 3 sec. time Delay Activity During ODR Task

prestimulus - 1 sec. cue sec. delay - 3 sec. time go signal rewarded saccade location Delay Activity During ODR Task

prestimulus - 1 sec. cue sec. delay - 3 sec. time go signal rewarded saccade location prestimulus cue delay preferred trials (adapted from Meyer et al. 2007) ‏ Delay Activity During ODR Task

prestimulus - 1 sec. cue sec. delay - 3 sec. time go signal rewarded saccade location non-preferred trials (adapted from Meyer et al. 2007) ‏ prestimulus cue delay Delay Activity during ODR task

Persistent Activity During an Oculom otor Delayed Response Task Funahashi, 2006

The Ring Model: Architecture  N E excitatory neurons; N I inhibitory neurons on a 1-D network with periodic boundary conditions (ring). Each neuron is characterized 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|). All to all connectivity : J(x-y)=J 0  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 = - I L – voltage gated channel currents + I syn + I stim + equations for gating variables of active channels e.g Na, K... I syn : synaptic current from other neurons in the network I stim : external noisy stimulus 2- Synaptic current: I syn = -g s(t) ( V – V syn ) After a presynaptic spike at t* : s(t) ---> s(t)+ exp[(-t +t*)/  syn ] for t > t*

Conductance Based Model Response of a Single Neuron to a Step of Current 100 ms 20mV

Conductance-Based Ring Model 200 msec Homogeneous Asynchronous State Stationary Bump time Neuron position

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. -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  Effective coupling: J(x)= J EE (x) - J EI (x) = J IE (x) - J II (x) J EE (x)= J IE (x) J II (x)= J EI (x)  E =  I =  I E = I I m E (x,t) = m I (x,t) = m(x,t)

J(x)= J 0 + J 1 cos (x)  (h) threshold linear i.e  (h) = h if h >0 and zero otherwise The Reduced Ring Model

The Phase Diagram of the Reduced Ring Model for an homogeneous external input I E = I I independent of x m(x,t)=m 0 J(x)= J 0 + J 1 cos (x)

Conductance-Based Ring Model 200 msec Stationary Uniform = Homogeneous Asynchronous State Stationary Bump time Neuron position

Reduced Ring Model with Delays Minimal rate model with delays: J(x)= J 0 + J 1 cos (x)  (h) threshold linear Sources of delays in neuronal systems: -Synaptic dynamics -Spikes dynamics -Axonal propagation -Dendritic processing

Equations for the Order Parameters

The Phase Diagram of the Rate Model with Delays: D=0.1  «epileptic » Chaos

Instabilities of the Stationary Uniform State The stationary uniform state, m(x,t) = m 0, is a trivial solution of the dynamics. The dispersion relation for the stability of this state is: = -1 + J n exp(-  D) For J(y)=J 0 +J 1 cos(x) there are 4 types of instabilities: 1- Rate instability (  =0, n=0) for J 0 =1 2-Turing instability (  =0, n=1) for J 1 =2 3-Hopf instability (  > 0, n=0) for J 0 cos(  D)=1 with  = -tan (  D) 4-Turing-Hopf instability (  > 0, n=1) for J 1 cos(  D)=1 with  solution of  =-tan(  D) With:

Instability of the Homogeneous Fixed Point to Homogeneous Oscillations If J 0 is sufficiently negative the stationary homogeneous fixed point undergoes a Hopf bifurcation with a spatially homogeneous unstable mode. D/  <<1 : bifurcation at J0 ~ -  /2  /D ; frequency of unstable mode is : f ~ ¼  /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+… Homogebneous synchronous oscillations driven by strong delayed inhibition t/  D/  m(t)

Homogeneous Oscillations in the Conductance Based Model Population Average Voltage Population oscillations in the  range induced by strong mutual inhibition Spikes are weakly synchronized ≠ spike to spike synchrony of type I neurons coupled via weak inhibition

The Homogeneous Oscillatory State and its Stability The homogeneous limit cycle of the homogeneous oscillation can be explicitely constructed: Step 1: 0<t < T 1 : I tot < 0 and m(t) ~ exp(-t/t); T 1 defined by I tot (T 1 )=0+ Step 2: T 1 < t < T 1 +D: m(t) satisfies: t dm/dt = -m + I tot (t-D)  m(t) = A exp(-t/t) + particular solution driven by the value of m in the previous epoch 0<t< T 1 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 I tot (T)=0. Stability can be studied 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

Results of the Stability Analysis of the Homogeneous Oscillatory State There are in general two Floquet exponents; e.g. assuming T<2 D :   0 =1: corresponds to the time translation invariance on the homogeneous limit cycle   1 corresponds to the spatially modulated mode cos(x) Stability iff |  1 [< 1.  1 =-1  period doubling instability  standing waves  1 =1  2 lines. In particular  oscillating bump NOTE: Numerical simulations show that these instabilities are subcritical ! With R=T-T 1 < D. This can be extended for arbitrary T.

The Standing Waves (at  1 =-1) For sufficiently strong modulated inhibition standing waves are found Rate Model Conductance-Based Model 25 ms

Bump of Synchronous Oscillatory Activity (at  1 =1) Rate Model 200 ms Conductance-Based Model

The Stationary Bump and its Instabilities -Like for D=0: The stationary uniform states looses stability via a Turing instability when J 1 crosses 2 from below. The resulting state is a stationary bump (SB). Instabilities: -Strong local excitation  rate instability  Neurons go to saturation -Strong inhibition  oscillatory instability  Localized synchronous oscillatory activity

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

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

Transition to Chaos in the Rate Model: N=100 Chaos

The Maximal Lyapunov Exponent J1J1 The lyapunov spectrum is computed numerically by integrating the linearized mean-field dynamics (three delayed coupled differential equations for the three order parameters describing the dynamics for N infinity) J0=-100

Chaos Emerges Via Period-Doubling Homogeneous oscillations: Amplitude does Not depend on J1 Standing Waves

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

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

End of Part 1