Nonlinear Dynamics of Cellular and Network Excitability and Oscillations

Transcription

1 Nonlinear Dynamics of Cellular and Network Excitability and Oscillations John Rinzel Computational Modeling of Neuronal Systems Fall 2007 Wilson-Cowan ( 72) rate model for excitatory/inhibitory popul ns. Firing rate model for binocular rivalry competing inhibitory popul ns + adaptation. Rhythms in developing spinal cord -- excitatory popul n, + synaptic depression. See review chapters: JR w/ B Ermentrout, 1998 Borisyuk w/ JR, 2005

2 What are τ e and τ i? JR: recruitment time constants for the network. Firing rate model (Amari-Wilson-Cowan) for dynamics of excitatory-inhibitory populations. τ e dr e /dt = -r e + S e (a ee r e -a ei r i + I e ) τ i dr i /dt = -r i + S i (a ie r e a ii r i + I i ) r i (t), r e (t) -- average firing rate (across population and over spikes ) τ e, τ i -- recruitment time scale S e (input), S i (input) input/output relations typically sigmoidal: S(x)=1/[1+exp((θ-x)/k)] where θ threshold ; k -- steepness (steeper for smaller k). a ee etc synaptic weights Network has no spatial structure; cells are random and sparsely connected. Wilson HR, JD Cowan. Excitatory and inhibitory interactions in localized populations of model neurons. Biophys J 12:1-22, 1972.

3 Wilson-Cowan Model dynamics in the phase plane. I e Phase plane, nullclines for range of J e.

4 Oscillator death r e Subthreshold I e Regime of repetitive activity

5 Oscillator Death but cells are firing Frequency Type II Type I I e

6 Dynamics of Perceptual Bistability Mutual inhibition with slow adaptation alternating dominance and suppression w/ N Rubin, A Shpiro, R Curtu, R Moreno

7 Idealized Models for Binocular Rivalry Two mutually inhibitory populations, corresponding to each percept. Firing rate model: r 1 (t), r 2 (t) Slow negative feedback: adaptation or synaptic depression. r 1 r 1 r 2 r 2 Slow adaptation, a 1 (t) Wilson; Laing and Chow τ dr 1 /dt = -r 1 + f(-βr 2 - φ a 1 + I 1 ) τ a da 1 /dt = -a 1 + f a (r 1 ) τ dr 2 /dt = -r 2 + f(-βr 1 - φ a 2 + I 2 ) τ a da 2 /dt = -a 2 + f a (r 2 ) τ a >> τ, f(u)=1/(1+exp[(θ-u)/k]) w/ N Rubin, A Shpiro, R Curtu, R Moreno

8 firing rate r 1 a 1 time Period, ms β = 1.1 β = 0.9 β = input strength I 1= I 2 Fast variables: r 1, r 2 phase plane changes slowly, τ a r 1 - nullcline r 2 - nullcline r 2 r 1

9 Modeling the rhythmic dynamics of developing spinal cord J Rinzel, NYU Background - network depression An ad hoc firing rate model Predictions/confirmations A specific mechanism for depression A minimal cell-based network model Mean field model? J Tabak, M O Donovan, C Marchetti, B Vladimirski

10 Interneurons Retrogradely Labeled By Calcium Dyes Injected Into The Ventrolateral Funiculus (VLF) Are Rhythmically Active (A.Ritter, P.Wenner, S.Ho) ventral Inject Calcium Dye caudal dorsal VLF LS1 Optical signals during rhythmic activity rostral

11 Synaptic Potentials Are Transiently Depressed After An Episode Of Spontaneous Activity 200 ms Normalized Amplitude episode episode Time (minutes)

12 Basic Model: mutual excitatory network. Activity level, a(t) ; availability of synaptic input, n τ a a = a (input)-a τ a a = a (n.a)-a

13 Spontaneous Rhythmic Episodes Can Be Modeled By An Excitatory Network With Fast And Slow Depression J Neuroscience, Network behavior is modeled by three variables Network Activity Activity-dependent Fast depression Activity-dependent Slow depression τ a a = a (s.d.a)-a τ d d = d (a)-d τ s s = s (a)-s Delayed Fast Depression Leads to Oscillations Fast and Slow Depression Produces Episodic Oscillations Activity (a) Slow Depression (s) 0.2 Fast Depression (d) time time

14 s Cycling dynamics: a-d phase plane, s-fixed. d a a Prescribed slow back-and-forth s(t) looks like episodes, but it s not autonomous.

15 FAST/SLOW Analysis a-d phase plane, s fixed. Bistable: upper OSC, lower SS a OSC: cycles SS: silent phase s syn s depressing Full system: s, slow depression syn s recovering

16 Fast/Slow Dissection -- a-s phase plane Kick out of Silent Phase => shorter Active Phase.

17 THEORY EXPERIMENT J Neuroscience, 2001.

18 MORE.? NMDA blocker

19 Reduction Of Connectivity Mimics The Recovery Of Activity During Excitatory Blockade τ a a = a (n.s.d.a)-a Experiment AP5 Model Decrease Connectivity Interval (mins) Interval Time (mins) Time

20 Phase space interpretation for effect of reduced connectivity τ a a = a (n.s.d.a)-a n, effective connectivity Interval time

21 MORE.? LIF cell-based network

22 N-cell I&F network. w/ J Tabak, B Vladimirski All-to-all excitatory coupling. Non-identical cells: I min < I j < I max If I max >1 some cells fire spontaneously. Slow depression, no cycles.

23 N-cell I&F network; N=50; Imin=0; Imax=1.1 gsyn eff raster plot s j vs j

24 γ, fraction of OSC cells. EXC I=0 I=1 OSC

25 Fast/slow analysis. Fast spiking dynamics, slow depression. asynchronous spiking total synaptic conductance is constant for fast dynamics, s j frozen cells are firing periodically, frequency r j =r(g syn,i j ) g j (t) replaced by temporal average Self-consistency eqn replaces fast dynamics: Slow dynamics: s j = α s (1-s j ) <β s,j > s j where <β s,j > depends on g syn and j Does self-consistency eqn lead to multi-branched surface, given {s j }?

26 Demonstration of bistability: Use self-consistency at each t, with current values of {s j } find 3 states: L, M, U U-M coalesce at end of Silent Phase L-M at end of Active Phase

27 MORE.? GABA A -V syn : cloride dynamics

28 w/ C Marchetti, J Tabak, M O Donovan J Neurosci 2005 V i < V syn ==> [Cl - ] int ==> V syn during activity V rest V Th V syn 0 V

29 Changes in [Cl - ] int and E GABAA are responsible, in part, for the postepisode depression of evoked currents (Chub & O Donovan) Intracellular current episode 500 µa Isoguvacine puffs 2 min Chloride concentration (mm) Chloride Flux GABA equilibrium Potential

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31 FAST/SLOW Analysis V mv V_Cl Cl influx Cl efflux

32 Ad hoc Mean Field model: SUMMARY Episodic rhythms - generated from mutual excitation with slow synaptic depression; - fast depression yields cycling during episodes. Bistability of fast subsystem leads to predictions for correlations between SP duration and next AP duration. Prediction/test for effect of reduced connectivity. GABA A synapses are functionally excitatory, ie V < V Cl ; V Cl is depolarized and it oscillates together with episodes; minimal model shows it. N-cell network model: Cycling during AP? Stochastic synapses for initiating next AP? Dynamics of heterogeneous network.