On the master equation approach to stochastic neurodynamics Paul C Bressloff Mathematical Institute and OCCAM, University of Oxford January 18, 2010 Paul C Bressloff On the master equation approach to stochastic neurodynamics
Part I. Master equation Paul C Bressloff On the master equation approach to stochastic neurodynamics
Master equation I Consider M homogeneous networks labelled k = 1 , . . . M , each containing N identical neurons Assume that each neuron can be in one of two states, quiescent or active ie. in the process of generating an action potential. Suppose that in the interval [ t , t + ∆ t ) there are n k ( t ) active neurons in the k th population Define population activity in terms of the fraction of active neurons according to ( t ) = n k ( t ) u ( N ) . k N Paul C Bressloff On the master equation approach to stochastic neurodynamics
Master equation II n1 ni n2 α F(Xi) ni+1 n3 Xi = N-1 Σ j wijnj Treat the number of active neurons n k ( t ) as a stochastic variable that evolves according to a one–step jump Markov process Rates of state transitions n k → n k ± 1 are chosen so that in the thermodynamic limit N → ∞ one obtains the deterministic Wilson-Cowan equations – transition rates not unique! Paul C Bressloff On the master equation approach to stochastic neurodynamics
Master equation III Let P ( n , t ) with n = ( n 1 , . . . , n N ) denote probability that m i ( t ) = n i for all i Probability distribution evolves according to birth-death master equation X N X dP ( n , t ) ˆ ˜ = T k , r ( n k , r ) P ( n k , r , t ) − T k , r ( n ) P ( n , t ) dt k =1 r = ± 1 where n k , r = ( n 1 , . . . , n k − 1 , n k − r , n k +1 , . . . , n N ). Master equation is supplemented by the boundary conditions P ( n , t ) ≡ 0 if n i = N + 1 or n i = − 1 for some i . Transition rates are X ! T k , − 1 ( n ) = α n k , T k , +1 ( n ) = NF w kl n l / N l Paul C Bressloff On the master equation approach to stochastic neurodynamics
Mean–field approximation Multiply both sides of master equation by n k and sum over all states n . This gives X d dt � n k � = r � T k , r ( n ) � r = ± 1 where � f ( n ) � = P n P ( n , t ) f ( n ) for any function of state f ( n ). Assume all statistical correlations can be neglected so that � T k , r ( n ) � ≈ T k , r ( � n � ) Setting u k = N − 1 � n k � leads to mean–field equation d dt u k = N − 1 [ T k , + ( N u ) − T k , − ( N u )] X ! = − α u k + F ≡ H k ( u ) w kl u l l Paul C Bressloff On the master equation approach to stochastic neurodynamics
Some comments (Perron–Frobenius theorem): For an irreducible weight matrix w , the transition 1 matrix of the master equation is also irreducible = ⇒ there exists a unique globally stable stationary solution. The nonlinear mean-field (rate) equations will generally have multiple attractors – 2 multistability. Convergence of P ( n , t ) to a unique steady state implies that fluctuations induce 3 transitions between basins of attraction of fixed points of mean-field equations. For large N the transition rates ∼ e − N τ for some τ – metastability Buice and Cowan use a different scaling: they interpret u k in rate equation as the 4 number rather than fraction of active neurons in k th population and take transition rates to be X ! T k , − 1 ( n ) = α n k , T k , +1 ( n ) = F w kl n l l Paul C Bressloff On the master equation approach to stochastic neurodynamics
Effects of fluctuations Away from critical points can analyze effects of fluctuations using a Van Kampen 1 system-size expansion in N − 1 Linear-noise (Gaussian) approximation – Gaussian fluctuations about a 2 deterministic trajectory of mean field equations System–size expansion to higher orders in N − 1 generates correction to MFT in 3 which the mean field couples to higher–order moments – equivalent to carrying out a loop expansion of the Buice–Cowan path integral (PCB SIAM 2009) The system size expansion breaks down close to critical points eg. where a fixed 4 point of mean-field equation becomes marginally stable – critical slowing down System-size expansion cannot account for exponentially small transition rates 5 between metastable states Alternative approach is to use a WKB approximation and analyze the effects of 6 fluctuations for large N using an effective Hamiltonian dynamical system Paul C Bressloff On the master equation approach to stochastic neurodynamics
Part II. Linear–noise (Gaussian) approximation Paul C Bressloff On the master equation approach to stochastic neurodynamics
Linear–noise approximation Perform the change of variables √ ( n k , t ) → ( Nu k ( t ) + N ξ k , t ) where u ( t ) is a solution of the mean–field WC equations Then ξ k ( t ) satisfies the Langevin equation X d ξ k = A kl ( u ( t )) ξ l + η k ( t ) , dt l where η k ( t ) represents a Gaussian process with zero mean and correlation function � η k ( t ) η l ( t ′ ) � = B k ( u ( t )) δ k , l δ ( t − t ′ ) and A kl ( u ) = ∂ H k ( u ) ∂ u l X ! B k ( u ) = N − 1 [ T k , + ( N u ) + T k , − ( N u )] = α u k + F w kl u l l Paul C Bressloff On the master equation approach to stochastic neurodynamics
Fokker–Planck equation Probability density P ( ξ, t ) satisfies the FP equation X X B k ( u ( t )) ∂ 2 ∂ P A kl ( u ( t )) ∂ [ ξ l P ( ξ , t )] + 1 = − P ( ξ , t ) ∂ξ 2 ∂ t ∂ξ k 2 k k , l k = E , I Solution is given given by the multidimensional Gaussian 0 1 X 1 @ − ξ k C − 1 A P ( ξ , t ) = p exp kl ( t ) ξ l (2 π ) M det C ( t ) k , l with the covariance matrix satisfying the equation ∂ C ∂ t = A ( u ) C + CA ( u ) T + B ( u ) . Paul C Bressloff On the master equation approach to stochastic neurodynamics
Deterministic E-I network Consider an E-I network given by the pair of Wilson-Cowan equations du E = − u E + F ( w EE u E − w EI u I + h E ) dt du I dt = − u I + F ( w IE u E − w II u I + h I ) , hI wIE wEE E I wII wEI hE Let ( ν ∗ E , ν ∗ I ) denote a fixed point with associated Jacobian „ − 1 + w EE ν ∗ « E (1 − ν ∗ − w EI ν ∗ E (1 − ν ∗ E ) E ) J = α w IE ν ∗ I (1 − ν ∗ − 1 − w II ν ∗ I (1 − ν ∗ I ) I ) ′ = F (1 − F ) assuming F is a sigmoid with F Paul C Bressloff On the master equation approach to stochastic neurodynamics
Phase diagram in the ( h E , h I ) –plane for a fixed weight matrix w Fixed point will be stable provided that the eigenvalues λ ± of J have negative real parts „ « q λ ± = 1 [Tr J ] 2 − 4Det J Tr J ± . 2 This leads to the stability conditions Tr J < 0 and Det J > 0. 0 Fold h I Hopf -5 Fold -10 -6 -4 -2 0 2 4 6 8 h E Hopf bifurcation curves satisfy Tr = 0 with Det J > 0. Saddle–node bifurcation curves given by the condition Det J = 0 with Tr J < 0. Paul C Bressloff On the master equation approach to stochastic neurodynamics
Power spectrum for stochastic E-I network Taking Fourier transforms of Langevin equation in linear noise approximation X − i ω e A kl e ξ k ( ω ) = ξ l ( ω ) + e η k ( ω ) . l = E , I This can be rearranged to give X e Φ − 1 ξ k ( ω ) = kl ( ω ) e η l ( ω ) , Φ kl ( ω ) = − i ωδ k , l − A kl . l = E , I The Fourier transform of the Gaussian process η k ( t ) has the correlation function η l ( ω ) � = 2 π B k ( u ∗ ) δ k , l δ ( ω + ω ′ ) . � e η k ( ω ) e The power spectrum is defined according to 2 πδ (0) P k ( ω ) = �| e ξ k ( ω ) | 2 � , so that X | Φ − 1 kl ( ω ) | 2 B l ( u ∗ ) . P k ( ω ) = l = E . I Paul C Bressloff On the master equation approach to stochastic neurodynamics
Noise-amplification of oscillations in an E-I network Power spectrum can be evaluated explicitly to yield P k ( ω ) = β k + γ k ω 2 | D ( ω ) | 2 with D ( ω ) = Det Φ( ω ) = − ω 2 + i ω Tr J + Det J and β E = J 2 22 B E ( u ∗ ) + J 2 12 B I ( u ∗ ) , γ E = B E ( u ∗ ) β I = J 2 21 B E ( u ∗ ) + J 2 11 B I ( u ∗ ) , γ I = B I ( u ∗ ) . Evaluating the denominator and using the stability conditions Tr J < 0 , Det J > 0, β k + γ k ω 2 P k ( ω ) = ( ω 2 − Ω 2 0 ) 2 + Γ 2 ω 2 where Γ = Tr J and Ω 2 0 = Det J . Paul C Bressloff On the master equation approach to stochastic neurodynamics
Peak in power spectrum (PCB 2010) 10 1 0.5 0.45 ν E 10 0 0.4 0.35 10 -1 P( ω ) 0.3 ν I 0.25 10 -2 0.2 0.15 10 -3 0.1 0 2 4 6 8 5 10 15 20 25 30 35 40 45 50 ω time t Suppose that the fixed point of mean-field equation is a focus (damped oscillations) Find that there is a resonance in the power spectrum around the subthreshold frequency Analogous to previous studies of noise-amplification of biochemical oscillations in cells (McKane et al 2007). Paul C Bressloff On the master equation approach to stochastic neurodynamics
Part III. WKB approximation and rare event statistics Paul C Bressloff On the master equation approach to stochastic neurodynamics
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