Stochastic dynamics of spiking neuron models and implications for network dynamics Nicolas Brunel
The question • What is the input-output relationship of single neurons?
The question • What is the input-output relationship of single neurons? • Given a (time-dependent) input, and a given statistics of the noise, what is the instantaneous firing rate of a neuron? I syn ( t ) = µ ( t ) + Noise ⇒ ν ( t )?
The question • What is the input-output relationship of single neurons? • Given a (time-dependent) input, and a given statistics of the noise, what is the instantaneous firing rate of a neuron? I syn ( t ) = µ ( t ) + Noise ⇒ ν ( t )? • Simplest case: response to time-independent input µ ( t ) = µ 0 ⇒ ν ( t ) = ν 0
The question • What is the input-output relationship of single neurons? • Given a (time-dependent) input, and a given statistics of the noise, what is the instantaneous firing rate of a neuron? I syn ( t ) = µ ( t ) + Noise ⇒ ν ( t )? • Simplest case: response to time-independent input µ ( t ) = µ 0 ⇒ ν ( t ) = ν 0 • Next step: response to time-dependent inputs � t K ( t − u ) µ 1 ( u ) du + O ( ǫ 2 ) µ ( t ) = µ 0 + ǫµ 1 ( t ) ⇒ ν ( t ) = ν 0 + ǫ −∞
The question • What is the input-output relationship of single neurons? • Given a (time-dependent) input, and a given statistics of the noise, what is the instantaneous firing rate of a neuron? I syn ( t ) = µ ( t ) + Noise ⇒ ν ( t )? • Simplest case: response to time-independent input µ ( t ) = µ 0 ⇒ ν ( t ) = ν 0 • Next step: response to time-dependent inputs � t K ( t − u ) µ 1 ( u ) du + O ( ǫ 2 ) µ ( t ) = µ 0 + ǫµ 1 ( t ) ⇒ ν ( t ) = ν 0 + ǫ −∞ • Fourier transform: response to sinusoidal inputs ν 1 ( ω ) = ˜ µ 1 ( ω ) ⇒ K ( ω ) µ 1 ( ω ) Of particular interest: high frequency limit (tells us how fast a neuron instantaneous firing rate can react to time-dependent inputs)
Noisy input current (mV) A 60 40 20 0 -20 0 20 40 60 B Spikes 0 20 40 60 40 C Firing rate (Hz) 30 20 10 0 0 20 40 60 t (ms)
How to compute the instantaneous firing rate • Consider a LIF neuron with deterministic + white noise inputs, τ m ˙ V = − V + µ ( t ) + σ ( t ) η ( t ) • 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 )] ∂t 2 ∂V 2 • Boundary conditions ⇒ links P and instantaneous firing probability ν – At threshold V t : absorbing b.c. + probability flux at V t = firing probability ν ( t ) : ∂P ∂V ( V t , t ) = − 2 ν ( t ) τ m 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 − r , t ) , ∂V ( V − σ 2 ( t )
How to compute the instantaneous firing rate • Consider a LIF neuron with deterministic + white noise inputs, τ m ˙ V = − V + µ ( t ) + σ ( t ) η ( t ) • 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 )] ∂t 2 ∂V 2 • Boundary conditions ⇒ links P and instantaneous firing probability ν – At threshold V t : absorbing b.c. + probability flux at V t = firing probability ν ( t ) : ∂P ∂V ( V t , t ) = − 2 ν ( t ) τ m 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 − r , t ) , ∂V ( V − σ 2 ( t ) ⇒ Time independent solution P 0 ( V ) , ν 0 ; ⇒ Linear response P 1 ( ω, V ) , ν 1 ( ω ) .
LIF model • ν 1 ( ω ) can be computed analytically for all ω in the case of white noise; in low/high frequency limits in the case of colored noise with τ n ≪ τ m • Resonances at f = nν 0 for high rates and low noise; • Attenuation at high f � ν 0 (white noise) σ √ ωτ m Gain ∼ � τ s ν 0 (colored noise) σ τ m • Phase lag at high f � π (white noise) 4 Lag ∼ 0 (colored noise) Brunel and Hakim 1999; Brunel et al 2001; Lindner and Schimansky-Geier 2001; Fourcaud and Brunel 2002
Spike generation: exponential integrate-and-fire • EIF: exponential integrate-and-fire neuron C dV = − g L ( V − V L ) + ψ ( V ) + I syn ( t ) dt � V − V T � ψ ( V ) = g L ∆ T exp ∆ T • Captures quantitatively very well the dynamics of a Hodgkin-Huxley-type neuron (Wang-Buszaki). • This is because activation curve of sodium currents can be well fitted by an exponential close to firing threshold. Fourcaud-Trocm´ e et al 2003
EIF vs cortical pyramidal cells - I-V curve dV F ( V ) + I in ( t ) = dt C � V − V T 1 � �� F ( V ) = E m − V + ∆ T exp τ m ∆ T Badel et al 2008
EIF - dynamical response A-2 B-2 • ν 1 ( ω ) can be computed in low/high Modulation amplitude, ν 1 /Ι 1 high rate, high noise 0 -1 10 frequency limits Phase shift, φ o -45 • In the high frequency limit, -2 10 o -90 ν 0 ν 0 =33Hz, σ =12.7mV ν 0 =33Hz, σ =12.7mV | ν 1 ( ω ) | ∼ -3 10 ∆ T ωτ m 0 1 2 3 0 1 2 10 10 10 10 10 10 10 10 Input frequency, f (Hz) Input frequency, f (Hz) φ ( ω ) ∼ π/ 2 B-3 A-3 Modulation amplitude, ν 1 /Ι 1 high rate, 0 -1 10 low noise Phase shift, φ • The whole function ν 1 ( ω ) can o -45 -2 10 be computed numerically using a o -90 method introduced by Richardson ν 0 =38Hz, σ =1.6mV ν 0 =38Hz, σ =1.6mV 2 ν 0 -3 10 ν 0 0 1 2 3 0 1 2 (2007) 10 10 10 10 10 10 10 10 Input frequency, f (Hz) Input frequency, f (Hz) Fourcaud-Trocm´ e et al 2003, Richardson 2007
Summary of high frequency behaviors Model Exponent Phase lag α φ ( f → ∞ ) 0 0 LIF, colored noise 0 . 5 45 ◦ LIF, white noise 1 90 ◦ EIF 2 180 ◦ QIF
Response of cortical pyramidal cells Boucsein et al 2009
Two variable models A second variable can be coupled to voltage to: • Include the effects of ionic currents which are activated below threshold (possibly leading to sub-threshold resonance): RF, GIF, aQIF , aEIF ⇒ Linear response can be computed as an expansion in ratio of time scales (Richardson et al 2003, Brunel et al 2003) • Include firing rate adaptation: aLIF , aQIF , aEIF • Include the effects of currents leading to bursting: IFB, aQIF , aEIF • Include a second compartment (soma + dendrite) ⇒ What are the effects of the second variable on the firing rate dynamics (linear response)?
Two compartmental model of a Purkinje cell • Can be fitted by two-compartment model dV s C s = − g s V s + g j ( V d − V s ) + I s dt dV d C d = − g d V d + g j ( V s − V d ) + I d dt
Two compartmental model of a Purkinje cell • or equivalently dV s τ s = − V s + γW + I s dt dW τ d = − W + V + I d dt • Fits give τ s ≪ 1 ms, τ d ∼ 5 ms (even though C s /g s = C d /g d ∼ 50 ms), γ ∼ 0 . 9 This is due to A s ≪ A d , g s ≪ g d ≪ g j
Linear firing rate response of 2C model • 2C model with exponential spike-generating current (2C-EIF) • Oscillatory input injected at the soma, noise at the dendrite • Very similar results obtained with multi-compartmental model based on a reconstructed PC with HH-type currents (Khaliq-Raman model)
Linear firing rate response: low frequency limit • When ωτ s ≪ 1 , the soma is driven instantaneously by dendritic + injected currents, V = γW + I 0 + I 1 e iωt . • Dynamics for the dendritic compartment becomes W = − (1 − γ ) W + I 0 + I 1 e iωt + σ √ τ d η ( t ) τ d ˙ with spikes occurring when W = ( V t − I 0 − I 1 e iωt ) /γ . • To recover a LIF model with constant threshold one can define X = W − V t − I 0 − I 1 e iωt γ and obtain τ d X = − X − V T I 0 I 1 σ τ d � ˙ γ (1 − γ )(1+ iωτ d ) e iωt + γ + γ (1 − γ )+ √ 1 − γ 1 − γ η ( t ) 1 − γ ⇒ ν 1 ∼ ν 1 ,LIF (1 + iωτ d ) ⇒ Amplitude of the response increases as a function of frequency
High frequency limit • At high frequency, response should be dominated by the spike-generating current as in the standard EIF . This should give ν 0 ν HF = 1 ∆ T iωτ s • Setting | ν LF | = | ν HF | we get 1 1 � 2 / 3 � γσ 1 √ ω ⋆ = τ 1 / 3 τ 2 / 3 2∆ T s d
Low and high frequency asymptotics vs simulations
Response of real Purkinje cells
Summary: single cell dynamics • High frequency behavior: controlled by spike generation dynamics • Low frequency behavior: determined by f-I curve • Intermediate frequencies: resonances can be due to various mechanisms: – Low noise regime: resonances at firing rate/harmonics (all models) – Subthreshold resonance can lead to firing rate resonance if noise is strong enough (Richardson et al 2003) – Specific spatial geometry of PC: high frequency resonance in response to somatic inputs • Linear response gives a good approximation of the dynamics as long as | ν 1 | < ν 0 • Cortical pyramidal cell response qualitatively well described by EIF; • Cerebellar Purkinje cell response qualitatively well described by 2-C EIF
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