The Ring Model of Cortical Dynamics: An overview

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

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

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

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

7. Orientation Columns and Hypercolumns in V1

8. + + + + Delay Activity During ODR Task prestimulus - 1 sec. time

9. + + + + + + + + Delay Activity During ODR Task prestimulus - 1 sec. cue -0.5 sec. time

10. + + + + + + + + + + + + Delay Activity During ODR Task prestimulus - 1 sec. cue -0.5 sec. delay - 3 sec. time

11. + + + + + + + + + + + + Delay Activity During ODR Task prestimulus - 1 sec. cue -0.5 sec. delay - 3 sec. go signal time + rewarded saccade location + + +

12. + + + + + + + + + + + + Delay Activity During ODR Task prestimulus - 1 sec. cue -0.5 sec. delay - 3 sec. go signal time + rewarded saccade location + + + preferred trials (adapted from Meyer et al. 2007)‏ cue delay prestimulus

13. + + + + + + + + + + + + Delay Activity during ODR task prestimulus - 1 sec. cue -0.5 sec. delay - 3 sec. go signal time + rewarded saccade location + + + non-preferred trials (adapted from Meyer et al. 2007)‏ prestimulus delay cue

14. Persistent Activity During an Oculomotor Delayed Response Task Funahashi, 2006

15. NE excitatory neurons; NI inhibitory neurons on a 1-D network withperiodic boundary conditions (ring). Each neuron is characterized by its position on the ring, x: - < x < . • The synaptic weight / connection probability from neuron (y,=E,I)to neuron (x, is a function of |x-y| and of,J (|x-y|). 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 The Ring Model: Architecture

16. Conductance-Based Ring Model 1- Dynamics: neurons described by Hodgkin-Huxley type model. C dV/dt = - IL– voltage gated channel currents + Isyn + Istim + equations for gating variables of 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*

17. 20mV 100 ms Conductance Based Model Response of a Single Neuron to a Step of Current

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

19. 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

20. mE (x,t) = mI(x,t) = m(x,t) Effective coupling: J(x)= JEE(x) - JEI(x) = JIE(x) - JII(x) 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

21. The Reduced Ring Model • J(x)= J0 + J1 cos (x) • F(h) threshold linear i.eF(h) = h if h >0 andzero otherwise

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

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

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

25. Equations for the Order Parameters

26. Chaos The Phase Diagram of the Rate Model with Delays: D=0.1 t «epileptic » 

27. 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 + Jnexp(-l D) With: • For J(y)=J0 +J1 cos(x) there are 4 types of instabilities: • 1- Rate instability (w=0, n=0) for J0=1 • 2-Turing instability (w=0, n=1) for J1=2 • 3-Hopf instability (w > 0, n=0) for J0 cos(w D)=1 with w = -tan (w D) • 4-Turing-Hopf instability (w > 0, n=1) for J1 cos(w D)=1 with w solution of w=-tan(w D)

28. Instability of the Homogeneous Fixed Point to Homogeneous Oscillations If J0 is sufficiently negative the stationary homogeneous fixed point undergoes a Hopf bifurcation with a spatially homogeneous 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+… Homogebneous synchronous oscillations driven by strong delayed inhibition D/t = 0.1 m(t) t/t

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

30. The Homogeneous Oscillatory State and its Stability The homogeneous limit cycle of the homogeneous oscillation 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 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

31. Results of the Stability Analysis of the Homogeneous Oscillatory State 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 modulated mode cos(x) Stability iff | b1 [< 1. • b1=-1  period doubling instability  standing waves • b1 =1 2 lines. In particular  oscillating bump • NOTE: Numerical simulations show that these instabilities are subcritical !

32. The Standing Waves (at b1=-1) For sufficiently strong modulated inhibition standing waves are found Rate Model Conductance-Based Model 25 ms

33. Bump of Synchronous Oscillatory Activity (at b1=1) Rate Model Conductance-Based Model 200 ms

34. 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). • Instabilities: • Strong local excitation  rate instability  Neurons go to saturation • -Strong inhibition  oscillatory instability Localized synchronous oscillatory activity

35. 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

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

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

38. J1 The Maximal Lyapunov Exponent 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

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

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

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

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

43. End of Part 1