Synchrony in Stochastic Pulse-coupled Neuronal Network Models Katie - PowerPoint PPT Presentation

Synchrony in Stochastic Pulse-coupled Neuronal Network Models Katie Newhall 1 c and Peter Kramer 1 Gregor Kovaˇ ciˇ Aaditya Rangan and David Cai 2 1 Rensselaer Polytechnic Institute, Troy, New York 2 Courant Institute, New York, New York January 19, 2009 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 ) = − g L ( v i ( t ) − V R ) + I i ( t ) , i = 1 , . . . , N dt δ ( t − t ik ) + S � � � δ ( t − ˜ I i ( t ) = f t jk ) N k j � = i k ! # ! " V R ≤ v i ( t ) < V T ◮ V R : reset potential (=0) ◮ V T : firing threshold (=1) g L : leakage rate (=1) ! % ! $ f and S : coupling strengths ( > 0)

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 = 1 , . . . , N, j � = i Reset: Membrane Potential: 1 0.9 0.8 0.7 Voltage 0.6 ! ! " # 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 Time

Classification by Mean External Driving Strength Mean voltage driven to V R + fν/g L without network coupling Superthreshold Subthreshold V R + fν/g L > V T V R + fν/g L < V T 1 1 0.9 0.9 0.8 0.8 0.7 0.7 voltage voltage 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 0 2 4 6 8 10 0 5 10 15 20 time time f = 0 . 01 , ν = 120 f = 0 . 01 , ν = 90 fν = 1 . 2 > 1 fν = 0 . 9 < 1

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 1 1 voltage voltage 0.5 0.5 0 0 0 2 4 6 8 10 0 2 4 6 8 10 Drift−Diffusion Process Driven Drift−Diffusion Process Driven 1 1 voltage voltage 0.5 0.5 0 0 0 2 4 6 8 10 0 2 4 6 8 10 time time f = 0 . 005 ≪ ( V T − V R ) /g L f = 0 . 1 = ∼ O (( V T − V R ) /g L ) ν = 2400 , fν = 1 . 2 , N = 1 ν = 12 , fν = 1 . 2 , N = 1

Types of Synchronous Firing: Partial Synchrony All-to-all connected network of N = 100 neurons 100 100 80 80 number of firing events neuron number 60 60 40 40 20 20 0 0 0 2 4 6 8 10 0 2 4 6 8 10 time time f = 0 . 01 , fν = 1 . 2 , S = 0 . 4

Types of Synchronous Firing: Imperfect Synchrony All-to-all connected network of N = 100 neurons 120 100 100 80 number of firing events 80 neuron number 60 60 40 40 20 20 0 0 0 2 4 6 8 10 0 2 4 6 8 10 time time f = 0 . 1 , fν = 1 . 2 , S = 2 . 0

Types of Synchronous Firing: Perfect Synchrony All-to-all connected network of N = 100 neurons Consistent total firing events; “synchronizable network” 120 100 100 80 number of firing events 80 neuron number 60 60 40 40 20 20 0 0 0 2 4 6 8 10 0 2 4 6 8 10 time time f = 0 . 001 , fν = 1 . 2 , S = 2 . 0

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?

(1) What is the probability to see repeated total firing events?

Model for Synchronous Firing Voltages of the other N − 1 neurons when the first neuron fires: Cascade-Susceptible Not Cascade-Susceptible 14 14 12 12 number of neurons number of neurons 10 10 8 8 6 6 4 4 2 2 0 0 0.5 0.6 0.7 0.8 0.9 1 0.5 0.6 0.7 0.8 0.9 1 voltage voltage f = 0 . 008 , fν = 1 . 0 f = 0 . 08 , fν = 1 . 0 S = 2 , N = 100 ( Bin size = S/N )

For What Parameters is Network Synchronizable? 14 Neuron will fire in a cascade if 12 instantaneous coupling from other number of neurons 10 firing neurons pushes its voltage over < 15 S/N 8 threshold 6 4 Probability all neurons included in 2 cascading event: P ( C ) 0 0.5 0.6 0.7 0.8 0.9 1 voltage Cascade-susceptibility described by event: N − 1 C k : V T − V ( k ) ≤ ( N − k ) S � C = C k where N k =1 C, C k ∈ [ V R , V T ] N and V (1) ≤ V (2) ≤ · · · ≤ V ( N − 1)

Computation of Cascade-Susceptibility Probability ◮ Condition upon time of first threshold crossing: � ∞ P ( C | T (1) = t ) p (1) P ( C ) = T ( t )d t. 0 ◮ Approximate condition: max neuron at threshold at time t P ( C | T (1) = t ) ≈ P ( C | V ( N ) ( t ) = V T ) ◮ Manipulate using elementary probability, P ( C | V ( N ) ( t ) = V T ) = 1 − P ( C c | V ( N ) ( t ) = V T ) N − 1 � P ( A j | V ( N ) ( t ) = V T ) = 1 − j =1 N − 1 where A k ≡ C c � ( C j ) k j = k +1 denotes event that cascade fails first at k th neuron

Computation of P ( A j | V ( N ) ( t ) = V T ) A 4 : V T ◮ Reduce to elementary combinatorial calculation of how neurons are distributed in bins (width S/N ), each independently distributed with  p v ( x, t ) for x ∈ [ V R , V T ] ,  � V T  V R p v ( x ′ , t ) d x ′ p v | V ( N ) ( x, t ) ≡  0 otherwise ,  ◮ Approximate single-neuron freely evolving voltage pdf p v ( x, t ) as Gaussian ◮ Sum probabilities of each configuration consistent with A j

Probability that Network is Cascade-Susceptible Larger fluctuations need larger coupling to maintain synchrony 1 1 N=1000 N=500 N=250 0.8 0.8 N=100 0.6 0.6 1 P(C) P(C) 0.4 0.4 P(C) f=0.0001 0.5 f=0.0005 f=0.001 0.2 0.2 f=0.002 0 0 5 10 f=0.004 S 0 0 0 0.01 0.02 0.03 0.04 0 0.01 0.02 0.03 S/N S/N fν = 1 . 2 , f = 0 . 001 fν = 1 . 2 , N = 100

Probability that Network is Cascade-Susceptible What about for networks where connections are not all-to-all? Synaptic failure: p f Sparsity: p c 1 1 0.8 0.8 0.6 0.6 P(C) P(C) 0.4 0.4 p f =0.00 p c =0.00 p c =0.25 p f =0.25 0.2 0.2 p f =0.50 p c =0.50 p c =0.75 p f =0.75 0 0 0 1 2 3 4 5 0 1 2 3 4 5 S S N = 100 , fν = 1 . 2 , f = 0 . 001 N = 100 , fν = 1 . 2 , N = 100 � (1 − p f ) Adjust P ( C ) by taking p v | V ( N ) ( x, t ) → p v | V ( N ) ( x, t ) (1 − p c )

Probability that Network is Cascade-Susceptible What about scale-free networks? Effect of local topology ◮ Distribution of connections: ◮ Consider higher order terms: P ( K = k ) ∝ k − 3 P ( k B ) ⇒ P ( k B | k A , A → B ) ◮ Account for topology in P ( A 2 ) : 1.0 $%&' 0.8 # # 0.6 $%&( P(C) ! ! 0.4 Thr.: m=30 " Sim.: m=30 " Thr.: m=50 0.2 Sim.: m=50 Thr.: m=100 "&) Sim.: m=100 0 0.02 0.04 0.06 0.08 S N = 4000 , fν = 1 . 2 , f = 0 . 001 red only neurons that fire (also with M. Shkarayev) blue neurons result of clustering neuron A connects to B

Probability that Network is Cascade-Susceptible What about non instantaneous synaptic coupling? P ( C ) = 0 Synaptic Time Course: H ( t ) t E e − t/τ E Synaptic Delay: T D τ 2 1000 1000 800 800 neuron number neuron number 600 600 400 400 200 200 0 0 2 4 6 8 10 0 0 2 4 6 8 10 time time N = 1000 , S = 0 . 1 , � T D � = 0 . 002 N = 1000 , S = 0 . 1 , τ E = 0 . 002 fν = 1 . 2 , f = 0 . 001 fν = 1 . 2 , f = 0 . 001 ,

(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 (1) T , of N independent neurons 100 80 Mean first passage time of N neurons neuron number � ∞ 60 tp (1) � t � = T ( t ) dt 0 40 p (1) T ( t ) = Np T ( t )(1 − F T ( t )) N − 1 20 0 0 2 4 6 8 10 time

Synchronous First Exit Problem 4 1 t=1.6 3.5 t=3.3 0.8 t=4.9 3 t=6.6 p v ( x, t ) 1 − F T ( t ) 2.5 0.6 2 0.4 1.5 1 0.2 0.5 0 0 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 voltage time ◮ Neuron reaches threshold - removed from system (absorbed) ◮ Probability neuron not fired is probability still in V R → V T � V T P ( T ≥ t ) = 1 − F T ( t ) = p v ( x, t ) dx V R ◮ p v ( x, t ) solves Kolmogorov Forward Equation (KFE)