Theoretical neuroscience: From single neuron to network dynamics - PowerPoint PPT Presentation

Theoretical neuroscience: From single neuron to network dynamics Nicolas Brunel

Outline • Single neuron stochastic dynamics • Network dynamics • Learning and memory

Single neurons in vivo seem highly stochastic

Single neuron stochastic dynamics: the LIF model • LIF neuron with deterministic + white noise inputs, dV dt = − V + µ ( t ) + σ ( t ) √ τ m η ( t ) τ m Spikes are emitted when V = V t , then neuron reset to V r ; • P ( V, t ) is described by Fokker-Planck equation = σ 2 ( t ) ∂ 2 P ( V, t ) ∂P ( V, t ) + ∂ τ m ∂V [( V − µ ( t )) P ( V, t )] ∂V 2 ∂t 2

Single neuron stochastic dynamics: the LIF model • P ( V, t ) is described by Fokker-Planck equation = σ 2 ( t ) ∂ 2 P ( V, t ) ∂P ( V, t ) + ∂ τ m ∂V [( V − µ ( t )) P ( V, t )] ∂V 2 ∂t 2 • Boundary conditions: – At threshold V t : absorbing b.c. + probability flux at V t = firing probability ν ( t ) : ∂V ( V t , t ) = − 2 ν ( t ) τ m ∂P P ( V t , t ) = 0 , σ 2 ( t ) – At reset potential V r : what comes out at V t must come back at V r ∂P r , t ) − ∂P r , t ) = − 2 ν ( t ) τ m r , t ) = P ( V + ∂V ( V + P ( V − ∂V ( V − r , t ) , σ 2 ( t )

LIF model: stationary inputs µ ( t ) = µ 0 , σ ( t ) = σ 0 Vt − µ 0 − ( V − µ 0 ) 2 � � � 2 ν 0 τ m σ exp( u 2 )Θ( u − V r ) du P 0 ( V ) = exp σ 2 σ V − µ 0 σ Vt − µ 0 1 � √ π σ exp( u 2 )[1 + erf ( u )] = τ m ν 0 Vr − µ 0 σ � x Vt − µ 0 � σ e x 2 dx e y 2 (1 + erf y ) 2 dy CV 2 2 πν 2 = 0 Vr − µ 0 −∞ σ

Time-dependent inputs • Given an arbitrary time-dependent input ( µ ( t ) , σ ( t ) ) what is the instantaneous firing rate ν ( t ) ?

Computing the linear firing rate response • Strategy: – start with small time-dependent perturbations around means, µ ( t ) = µ 0 + ǫµ 1 ( t ) , σ ( t ) = σ 0 + ǫσ 1 ( t ) – linearize FP equation and obtain the linear response of P = P 0 + ǫP 1 ( t ) and ν = ν 0 + ǫν 1 ( t ) (solution of inhomogeneous 2nd order ODE). � t R µ ( t − t ′ ) µ 1 ( t ′ ) + R σ ( t − t ′ ) σ 1 ( t ′ ) dt ′ ν 1 ( t ) = ν 1 ( ω ) ˜ = R µ ( ω )˜ µ 1 ( ω ) + R σ ˜ σ 1 ( ω ) – R µ and R σ can be computed explicitly in terms of confluent hypergeometric functions. – go to higher orders in ǫ ...

LIF model: linear rate response R µ ( ω ) (changes in µ ) √ 2 iωτ m ) • High frequency behavior: R µ ( ω ) ∼ ν 0 / ( σ 0 √ t initial response for step currents. • Translates into a

More realistic models High ω behavior • Colored noise inputs: dV � τ s τ m = − V + µ ( t ) + σ ( t ) W dt R µ ( ω ) ∼ dW τ m − W + √ τ m η ( t ) τ s = dt • More realistic spike generation: dV dt = − V + F ( V ) + µ ( t ) + σ ( t ) √ τ m η ( t ) τ m Spike emitted when V → ∞ ; then reset at V r – EIF: F ( V ) = ∆ t exp(( V − V T ) / ∆ t ) R µ ( ω ) 1 /ω ∼ – QIF: F ( V ) ∼ V 2 1 /ω 2 R µ ( ω ) ∼ – PIF: F ( V ) ∼ V α 1 /ω α/ ( α − 1) R µ ( ω ) ∼

Conclusions • In simple spiking neuron models, response of instantaneous firing rate can be much faster than the response of the membrane; • EIF model: fits well pyramidal cell data, allows to understand quantitatively factors controlling speed of firing rate response; • Cut-off frequency of real neurons is very high ( ∼ 200 Hz or higher) ⇒ allows very fast population response to time dependent inputs • EIF can be mapped to both LNP and Wilson-Cowan-type firing rate models, with a time constant that depends on intrinsic parameters of the cell, and on instantaneous rate itself

Local networks in cerebral cortex • Size ∼ cubic millimeter • Total number of cells ∼ 100,000 • Types of cells: – pyramidal cells - excitatory (80%) – interneurons - inhibitory (20%) • Connection probability ∼ 10% • Synapses/cell: ∼ 10,000 (total 10 9 synapses/mm 3 ) • Each synapse has a small effect: depo- larization/ hyperpolarization ∼ 1-10% of threshold.

Randomly connected network of LIFs • N neurons. Each neuron receives K < N randomly chosen connections from other neurons. Couplings between neurons J ( J < 0 is total coupling strength). • Neurons = leaky integrate-and-fire: dV i ( t ) τ m = − V i + I i dt Threshold V t , reset V r • Total input of a neuron i at time t √ τ m η i ( t ) � � S ( t − t k I i ( t ) = µ ext + J c ij j ) + σ ext j k where S ( t ) describes time course of PSCs, t k j spike time of k th spike of neuron j , c ij chosen randomly such that � j c ij = K for all i .

Analytical description of irregular state • If neurons are firing approximately as Poisson processes, and connection probability is small ( K/N ≪ 1 ), then the recurrent inputs to a neuron can be approximated as ext + J 2 Kν ( t − D ) τ √ τη i ( t ) � σ 2 I i ( t ) = µ ext + JKτν ( t − D ) + where η i ( t ) are uncorrelated white noise. • We can use again Fokker-Planck formalism, ∂t = σ 2 ( t ) τ ∂P ∂V 2 + ∂ ∂P ∂V [( V − µ ( t )) P ] , 2 where – µ ( t ) = average input (external − recurrent inhibitory) µ ( t ) = µ ext + JKτν ( t − D ) – σ ( t ) = ‘intrinsic’ noise due to recurrent interactions σ 2 ( t ) = σ 2 ext + J 2 Kν ( t − D ) τ

Asynchronous state, linear stability analysis 1. Asynchronous state (constant instantaneous firing rate): Vt − µ 0 1 � √ π σ 0 exp( u 2 )[1 + erf ( u )] = τ m ν 0 Vr − µ 0 σ 0 µ 0 = µ ext + KJν 0 τ m σ 2 σ 2 ext + KJ 2 ν 0 τ m = 0 2. Linear stability analysis: P ( V, t ) = P 0 ( V ) + ǫP 1 ( V, λ ) exp( λt ) ν 0 ( t ) = ν 0 + ǫν 1 ( λ ) exp( λt ) . . . ⇒ obtain eigenvalues λ 3. Instabilities of asynchronous state occur when Re ( λ ) = 0 ; 4. Weakly non-linear analysis: behavior beyond the bifurcation point 5. Finite size effects

Randomly connected E-I networks ✘ ✘ ✾ ❍❍❍❍❍❍❍ ❥ ✏✏✏✏✏✏✏✏✏✏✏ ✶ ❆ ❑ ❆

Conclusions - network dynamics • Network dynamics can be studied analytically using Fokker-Planck formalism; • Inhibition-dominated networks settle in highly irregular states, that can be either asynchronous or synchronous; • Such irregular states reproduce some of the main experimentally observed features of spontaneous activity in cortex in vivo: – Highly irregular firing of single cells at low rates; – Broad distribution of firing rates (close to lognormal) – Weak correlations between cells • Synchronous irregular oscillations similar to fast oscillations observed in cerebellum, hippocampus, cerebral cortex • LFP spectra from all these structures can be fitted quantitatively by the model • Irregularity persists in randomly connected networks in the absence of noise • Irregular dynamics can be truly chaotic (positive Lyapunov exponents) or ‘stably chaotic’ (negative Lyapunov exponents)

Synaptic plasticity, learning and memory

Synaptic plasticity and network dynamics: future challenges • So far, most studies of learning and memory in networks have focused on networks with fixed connectivity (typically Hebbian - assumed to be the result of learning) • With Hebbian connectivity matrices, networks become multistable - with one background state, and a multiplicity of ‘selective’ attractors representing stored memories. • Challenges: – Devise ‘learning rules’ (i.e. dynamical equations for synapses) consistent with known data – Insert such rules in networks, and study how inputs with prescribed statistics shape network attractor landscape – Study maximal storage capacity of the network, with different types of attractors – Learning rules that are able to reach maximal capacity?