Synchrony in Stochastic Pulse-coupled Neuronal Network Models Katie Newhall Gregor Kovačič and Peter Kramer Aaditya Rangan and David Cai 2 Rensselaer Polytechnic Institute, Troy, New York 2 Courant Institute, New York, New York January 9, 29 Stochastic Models in Neuroscience Marseille, France
Motivation and Goals Motivation: Neuronal Model Behavior Study the most basic point neuron models, network models, and coarse-grained models for use in computations Develop new analytical techniques Look for simple, universal network mechanisms Goals: Understand Oscillatory Dynamics of IF Model Generalize classical problem of perfectly synchronous Integrate-and-Fire (IF) network oscillations to stochastic setting Quantitative analysis of mechanism sustaining synchrony Lessons for understanding more complicated models and types of oscillations?
Outline Network dynamics: model and simulations Characterization of mechanism for sustaining stochastically driven synchronized dynamics Computation of probability of repeated total firing events Computation of firing rates based on first passage time First passage time calculation Approximation of first passage time
Excitatory Neuronal Network All-to-all connected, current based, integrate-and-fire (IF) network dv i (t) dt = g L (v i (t) V R ) + I i (t), i =,...,N I i (t) = f k δ(t t ik ) + S δ(t t jk ) N j i k v v3 v2 v4 V R v i (t) < V T V R : reset potential (=) V T : firing threshold (=) g L : leakage rate (=) f and S: coupling strengths (> )
Integrate & Fire Dynamics v i (t) evolves according to the voltage ODE until t ik, when v i ( t ik ) V T At t ik, v i (t) is reset to V R : v i ( t + ik ) = V R S/N is added to the voltage, v j, for j =,...,N,j i Reset: Membrane Potential: V R V T Voltage.9.8.7.6.5.4.3.2..5.5 2 Time
Classification by Mean External Driving Strength Mean voltage driven to V R + fν/g L without network coupling Superthreshold V R + fν/g L > V T Subthreshold V R + fν/g L < V T voltage.9.8.7.6.5.4.3.2. 2 4 6 8 time f =.,ν = 2 fν =.2 > voltage.9.8.7.6.5.4.3.2. 5 5 2 f =.,ν = 9 fν =.9 < time
Classification by Fluctuation Size Approximate Poisson point process with rate ν and amplitude f by drift-diffusion process with drift coef fν and diffusion coef f 2 ν/2 Diffusion Approx. Valid Diffusion Approx. Not Valid Poisson Process Driven Poisson Process Driven voltage.5 voltage.5 2 4 6 8 Drift Diffusion Process Driven 2 4 6 8 Drift Diffusion Process Driven voltage.5 2 4 6 8 time f =.5 (V T V R )/g L ν = 24,fν =.2,N = voltage.5 2 4 6 8 time f =. = O((V T V R )/g L ) ν = 2,fν =.2,N =
Types of Synchronous Firing: Partial Synchrony All-to-all connected network of N = neurons number of firing events 8 6 4 2 neuron number 8 6 4 2 2 4 6 8 time f =.,fν =.2,S =.4 2 4 6 8 time
Types of Synchronous Firing: Imperfect Synchrony All-to-all connected network of N = neurons 2 number of firing events 8 6 4 2 neuron number 8 6 4 2 2 4 6 8 time f =.,fν =.2,S = 2. 2 4 6 8 time
Types of Synchronous Firing: Perfect Synchrony All-to-all connected network of N = neurons Consistent total firing events; synchronizable network 2 number of firing events 8 6 4 2 neuron number 8 6 4 2 2 4 6 8 time f =.,fν =.2,S = 2. 2 4 6 8 time
Model for Self-Sustaining Synchrony Fluctuations in input desynchronize the network Between total firing events N neurons behave independently Pulse coupling synchronizes the network First neuron firing causes cascading event, pulling all other neurons over threshold due to synaptic coupling (S) After each total firing event, all neurons reset, process repeats Which systems are synchronous? What is the mean firing rate of the network?
() What is the probability to see repeated total firing events?
Model for Synchronous Firing Voltages of the other N neurons when the first neuron fires: Cascade-Susceptible Not Cascade-Susceptible 4 4 2 2 number of neurons 8 6 4 number of neurons 8 6 4 2 2.5.6.7.8.9 voltage f =.8,fν =..5.6.7.8.9 voltage f =.8,fν =. S = 2,N = (Bin size = S/N)
For What Parameters is Network Synchronizable? number of neurons 4 2 8 6 4 2 < 5 S/N.5.6.7.8.9 voltage Cascade-susceptibility described by event: C = N k= Neuron will fire in a cascade if instantaneous coupling from other firing neurons pushes its voltage over threshold Probability all neurons included in cascading event: P(C) C k where C k : V T V (k) (N k) S N C,C k [V R,V T ] N and V () V (2) V (N )
Computation of Cascade-Susceptibility Probability Condition upon time of first threshold crossing: P(C) = P(C T () = t)p () T (t)dt. Approximate condition: max neuron at threshold at time t P(C T () = t) P(C V (N) (t) = V T ) Manipulate using elementary probability, P(C V (N) (t) = V T ) = P(C c V (N) (t) = V T ) where A k C c k N j=k+ (C j ) N = P(A j V (N) (t) = V T ) j= denotes event that cascade fails first at kth neuron
Computation of P(A j V (N) (t) = V T ) A 4 : Reduce to elementary combinatorial calculation of how neurons are distributed in bins (width S/N), each independently distributed with p v (x,t) VT for x [V R,V T ], p v V (N)(x,t) V R p v (x,t)dx otherwise, Approximate single-neuron freely evolving voltage pdf p v (x,t) as Gaussian Sum probabilities of each configuration consistent with A j V T
Probability that Network is Cascade-Susceptible Larger fluctuations need larger coupling to maintain synchrony.8 N= N=5 N=25 N=.8 P(C).6.4 P(C).5.2 5 S..2.3.4 S/N fν =.2,f =. P(C).6.4 f=. f=.5.2 f=. f=.2 f=.4..2.3 S/N fν =.2,N =
Probability that Network is Cascade-Susceptible What about for networks where connections are not all-to-all? Synaptic failure: p f Sparsity: p c.8.8 P(C).6.4.2 p f =. p f =.25 p f =.5 p f =.75 2 3 4 5 S N =,fν =.2,f =. P(C).6.4.2 p c =. p c =.25 p c =.5 p c =.75 2 3 4 5 S N =,fν =.2,N = Adjust P(C) by taking p v V (N)(x,t) { ( pf ) ( p c ) p v V (N)(x,t)
Probability that Network is Cascade-Susceptible What about scale-free networks? Effect of local topology Distribution of connections: P(K = k) k 3 Account for topology in P(A 2 ): L K A=4 A B K =3 B L L=2 red only neurons that fire blue neurons result of clustering neuron A connects to B P(C) Consider higher order terms:..8.6.4.2 P(k B ) P(k B k A,A B).2.4.6.8 S Thr.: m=3 Sim.: m=3 Thr.: m=5 Sim.: m=5 Thr.: m= Sim.: m= N = 4,fν =.2,f =. (also with M. Shkarayev)
Probability that Network is Cascade-Susceptible What about non instantaneous synaptic coupling? P(C) = Synaptic Delay: T D Synaptic Time Course: H(t) t e t/τ τe 2 E 8 8 neuron number 6 4 neuron number 6 4 2 2 2 4 6 8 time N =,S =., T D =.2 fν =.2,f =. 2 4 6 8 time N =,S =.,τ E =.2 fν =.2,f =.,
(2) What is the mean time between total firing events?
Synchronous First Exit Problem N neurons - don t want to solve N-dimensional mean exit time problem! Obtain PDF of one neuron s exit time, p T (t), and find PDF of minimum exit time, p () T, of N independent neurons neuron number 8 6 4 2 Mean first passage time of N neurons t = tp () T (t)dt p () T (t) = Np T(t)( F T (t)) N 2 4 6 8 time
Synchronous First Exit Problem pv(x, t) 4 3.5 3 2.5 2.5.5 t=.6 t=3.3 t=4.9 t=6.6 FT(t).8.6.4.2.2.4.6.8 voltage 2 4 6 8 time Neuron reaches threshold - removed from system (absorbed) Probability neuron not fired is probability still in V R V T P(T t) = F T (t) = VT V R p v (x,t)dx p v (x,t) solves Kolmogorov Forward Equation (KFE)
Single Neuron Kolmogorov Forward Equation KFE: t p v(x,t) = x [g L(x V R )p v (x,t)] + ν [p v (x f,t) p v (x,t)] Taylor Expand KFE - Diffusion Approximation: t p v(x,t) = x [(g L(x V R ) fν)p v (x,t)] + f2 ν 2 2 x 2p v(x,t) Boundary Conditions: absorbing at V T : p v (V T,t) = reflecting at V R : J[p v ](V R,t) = Flux: J[p v ](x,t) = [(g L (x V R ) fν)p v (x,t)] f2 ν 2 x p v(x,t)
Solution to Single Neuron KFE Eigenfunction expansion: p v (x,t) = p s (x) A n Q n (x)e λnt n= Define: z(x) = g L(x V R ) fν f νg L Eigenfunctions are Confluent Hypergeometric Functions: ( ) ( ) c M λn 2g Q n (x) = L, 2,z2 (x) + c 2 U λn 2g L, 2,z2 (x) ( ) ( ) c 3 M λn 2g L, 2,z2 (x) + c 4 U λn 2g L, 2,z2 (x) for z(x) < for z(x) Stationary Solution: p s (x) = Ne z2 (x) Initial Conditions: p v (x,) = δ(x V R ) A n = Q n(v R ) R VT V p R s(x)q 2 n (x)dx
Synchronous Gain Curves dashed - Gaussian approx. solid - first passage time circles - simulations black - deterministic rate Good agreement where diffusion approximation valid For fixed (superthreshold) fν, dependence f (a) firing rate 2.5 m /ˆτ.5..2 f.5 f=.5 f=. f=.2 f=.5.5.5 fν N=, S=5.
Synchronous Gain Curves dashed - Gaussian approx. solid - first passage time circles - simulations For fixed (superthreshold) fν, dependence ln N (b) firing rate.8.6.4 N= N=5 N= m /ˆτ.45.2.35 6 8 ln(n).7.8.9..2 fν.4 f =.,S = 2.
Summary Synchronous firing over a wide range of parameters Balance between fluctuations and coupling strength Synchronous firing rate in terms of single neuron properties Driven by time first neuron crosses threshold Network properties are calculated accurately through exit time problems where diffusion approximation is valid This work was supported by a NSF graduate fellowship (Newhall & al. (2) Comm in Math Sci, Vol. 8, No. 2, pp. 54-6)