13 Continuity Equation and the Fokker-Planck Approach

13.5 Networks of Nonlinear Integrate-and-Fire Neurons

In Chapter 5 it was shown that nonlinear integrate-and-fire neurons such as the exponential integrate-and-fire model provide an excellent approximation to the dynamics of single neurons, much better than the standard leaky integrate-and-fire model. This section indicates how the Fokker-Planck approach can be used to analyze networks of such nonlinear integrate-and-fire models.

We determine, for arbitrary nonlinear integrate-and-fire models driven by a diffusive input with constant mean μ=RI0\mu=R\,I_{0} and noise σ2\sigma^{2}, the distribution of membrane potentials p0(u)p_{0}(u) as well as the linear response of the population activity

A(t)=A0+A1(t)=A0+0G(s)I1(t-s)ds,A(t)=A_{0}+A_{1}(t)=A_{0}+\int_{0}^{\infty}G(s)\,I_{1}(t-s)\,{\text{d}}s, (13.37)

to a drive μ(t)=R[I0+I1(t)]\mu(t)=R\,[I_{0}+I_{1}(t)] (434). As explained in Section 13.4, the knowledge of p0p_{0} is sufficient to predict, for coupled populations of integrate-and-fire neurons, the activity A0A_{0} in the stationary state of asynchronous firing. Moreover, we will see that the filter G(s)G(s) contains all the information need to analyze the stability of the stationary state of asynchronous firing. In this Section we focus on one-dimensional nonlinear integrate-and-fire models, and add adaptation later on in Section 13.6.

13.5.1 Steady state population activity

We consider a population of nonlinear integrate-and-fire models in the diffusion limit. We start with the continuity equation (13.7)

tp(u,t)=-uJ(u,t)+A(t)δ(u-ur)-A(t)δ(u-θreset).\qquad{\partial\over\partial t}p(u,t)=-{\partial\over\partial u}J(u,t)+A(t)\,% \delta(u-u_{r})-A(t)\,\delta(u-\theta_{\rm reset})\,. (13.38)

In the stationary state, the membrane potential density does not depend on time, p(u,t)=p0(u)p(u,t)=p_{0}(u), so that the left-hand side of (13.38) vanishes. Eq. (13.38) therefore simplifies to

uJ(u,t)=A(t)δ(u-ur)-A(t)δ(u-θreset).{\partial\over\partial u}J(u,t)=A(t)\,\delta(u-u_{r})-A(t)\,\delta(u-\theta_{% \rm reset})\,. (13.39)

Therefore the flux takes a constant value, except at two values: at the numerical threshold θreset\theta_{\rm reset}, at which the membrane potential is reset; and at the potential uru_{r}, to which the reset occurs. The flux vanishes for u>θresetu>\theta_{\rm reset} and for u<uru<u_{r}. For ur<u<θresetu_{r}<u<\theta_{\rm reset} the constant value J(u,t)=c>0J(u,t)=c>0 still needs to be determined.

A neuron in the population under consideration is driven by the action potentials emitted by presynaptic neurons from the same or other populations. For stochastic spike arrivals at rate νk\nu_{k}, where each spike causes a jump of the voltage by an amount wkw_{k}, the contributions to the flux have been determined in Eqs. (13.10) and (13.11). In the diffusion limit the flux according to Eq. (13.21) is

J(u,t)=1τm[f(u)+μ(t)-12σ2(t)u]p(u,t).J(u,t)={1\over\tau_{m}}\,[f(u)+\mu(t)-{1\over 2}\sigma^{2}(t){\partial\over% \partial u}]\,p(u,t)\,. (13.40)

In the stationary state, we have p(u,t)=p0(u)p(u,t)=p_{0}(u) and J(u,t)=cJ(u,t)=c for ur<u<θresetu_{r}<u<\theta_{\rm reset}. (For u<uru<u_{r} we have J(u,t)=0J(u,t)=0.) Hence Eq. (13.40) simplifies to a first-order differential equation

dp0(u)du=2τmσ2[f(u)+μτmp0(u)-c].{{\text{d}}p_{0}(u)\over{\text{d}}u}={2\tau_{m}\over\sigma^{2}}\left[{f(u)+\mu% \over\tau_{m}}p_{0}(u)-c\right]\,. (13.41)

The differential equation (13.41) can be integrated numerically starting at the upper bound u=θresetu=\theta_{\rm reset} with initial condition p0(θreset)=0p_{0}(\theta_{\rm reset})=0 and dp0/du|θreset=-1=-2cτm/σ2{\text{d}}p_{0}/{\text{d}}u|_{\theta_{\rm reset}}=-1=-2c\tau_{m}/\sigma^{2}. When the integration passes u=uru=u_{r} the constant switches from c to zero. The integration is stopped at a lower bound ulowu_{\rm low} when p0p_{0} has approached zero. The exact value of the lower bound is of little importance.

At the end of the integration, the surface under the voltage distribution ulowθresetp0(u)du=1\int_{u_{\rm low}}^{\theta_{\rm reset}}p_{0}(u){\text{d}}u=1 which determines the constant cc and therefore the firing rate A0A_{0} in the stationary state.

Example: Exponential integrate-and-fire model

The exponential integrate-and-fire model (156) as defined in Eq. (5.6) of Chapter 5 is characterized by the differential equation

τ   d      d   tu=-(u-urest)+ΔTexp(u-ϑrhΔT)+μ,\tau{{\text{d}}\over{\text{d}}t}u=-(u-u_{\rm rest})+\Delta_{T}\,\exp\left({u-% \vartheta_{rh}\over\Delta_{T}}\right)+\mu\,, (13.42)

where μ\mu is the total drive in units of the membrane potential. The result of the numerical integration of Eqs. (13.39) and (13.41) depends critically on the relative value of the reset potential, the driving potential μ\mu with respect to the reset uru_{r}, and the rheobase firing threshold ϑrh\vartheta_{rh}, as well as on the level σ\sigma of diffusive noise. Two representative scenarios are shown in Fig. 13.8A. Case (i) has very little noise (σ=2\sigma=2mV) and the total drive (μ=-45\mu=-45mV) is above the rheobase firing threshold (ϑrh=-53\vartheta_{rh}=-53mV). In this case, the membrane potential density a few mV below the reset potential (ur=-60u_{r}=-60mV) is negligible. Because of the strong drive, the model neuron model is in the super-threshold regime (cf. Section 8.3 in Ch. 8) and fires regularly with a rate of about 44Hz.

Case (ii) is different, because it has a larger amount of noise (σ=6\sigma=6mV) and a negligible drive of μ=ur=-60\mu=u_{r}=-60mV. Therefore, the distribution of membrane potentials is broad, with a maximum at uru_{r} and firing is noise driven and occurs at a low rate of 5.6Hz.

For both noise levels, the frequency-current curve A0=ν=gσ(I0)A_{0}=\nu=g_{\sigma}(I_{0}) is shown in Fig. 13.8B, as a function of the total the drive μ=RI\mu=R\,I. We emphasize that for very strong super-threshold drive, the population firing rate A0A_{0} of noisy neurons is lower than that of noise-free neurons.

Fig. 13.8: Exponential integrate-and-fire neurons. A. The stationary membrane potential density p0(u)p_{0}(u) in the regime of low-noise and super-threshold drive (case (i): σ=2\sigma=2mV, μ=-45\mu=-45mV >ϑrh=-53>\vartheta_{rh}=-53mV >ur=-60>u_{r}=-60mV) and in the regime of high noise and subthreshold drive (case (ii): σ=6\sigma=6mV, μ=ur=-60\mu=u_{r}=-60mV <ϑrh=-53<\vartheta_{rh}=-53mV). Note that the exact value of the numerical firing threshold (θreset=0\theta_{\rm reset}=0mV) is irrelevant because the membrane p0(u)p_{0}(u) is, for all potentials above -30-30mV, very small. B. The stationary population firing rate A0=gσ(I0)A_{0}=g_{\sigma}(I_{0}) as a function of the total drive μ=RI0\mu=RI_{0} of the population, in the regimes of high noise (solid line, σ=6\sigma=6mV), low noise (dashed line, σ=2\sigma=2mV), and in the absence of noise (dotted, σ=0\sigma=0). The two cases depicted in A are marked by open and filled symbols; adapted from Richardson (434).

13.5.2 Response to modulated input (*)

So far we restricted the discussion to a constant input I0I_{0}. We now add a small periodic perturbation

I(t)=I0+ϵcos(ωt),I(t)=I_{0}+\epsilon\,\cos(\omega\,t), (13.43)

where ω=2π/T\omega=2\pi/T; the parameter TT denotes the period of the modulation. We expect the periodic drive to lead to a small periodic change in the population activity

A(t)=A0+A1(ω)cos(ωt+ϕA(ω)).A(t)=A_{0}+A_{1}(\omega)\,\cos(\omega\,t+\phi_{A}(\omega))\,. (13.44)

Our aim is to calculate the complex number G^\hat{G} that characterizes the linear response or ‘gain’ at frequency ω\omega

G^(ω)=A1(ω)ϵeiϕA(ω).\hat{G}(\omega)={A_{1}(\omega)\over\epsilon}\,e^{i\phi_{A}(\omega)}\,. (13.45)

The result for a population of exponential integrate-and-fire neurons is shown in Fig. 13.9.

Once we have found the gain G^\hat{G} for arbitrary frequencies ω\omega, an inverse Fourier transform of G^\hat{G} gives the linear filter G(s)G(s). The linear response of the population activity A(t)=A0+ΔA(t)A(t)=A_{0}+\Delta A(t) to an arbitrary input current I(t)=I0+ΔI(t)I(t)=I_{0}+\Delta I(t) is

ΔA(t)=0G(s)ΔI(t-s)ds.\Delta A(t)=\int_{0}^{\infty}G(s)\,\Delta I(t-s)\,{\text{d}}s\,. (13.46)

A linear response is a valid description if the change is small:

ΔA(t)A(t).\Delta A(t)\ll A(t)\,. (13.47)

In our case of periodic stimulation, the amplitude of the input current is scaled by ϵ\epsilon. We are free to choose ϵ\epsilon small enough to fulfill condition (13.47).

The small periodic drive at frequency ω\omega leads to a small periodic change in the density of membrane potentials

p(u,t)=p0(u)+p1(u)cos(ωt+ϕp(u))p(u,t)=p_{0}(u)+p_{1}(u)\,\cos(\omega\,t+\phi_{p}(u)) (13.48)

which has a phase lead (ϕp>0\phi_{p}>0) or lag (ϕp<0\phi_{p}<0) with respect to the periodic drive. We assume that the modulation amplitude in the input current ϵ\epsilon is small so that p1(u)p0(u)p_{1}(u)\ll p_{0}(u) for most values of uu. We say that the change is at most of ‘order ϵ\epsilon’.

Similarly to the membrane potential density, the flux J(u,t)J(u,t) will also exhibit a small perturbation of order ϵ\epsilon. For the exponential integrate-and-fire model of Eq. (13.42) with urest=0u_{\rm rest}=0 the flux is (cf. Eq. (13.40))

J(u,t)=[-u+RI(t)τm+ΔTτmexp(u-ϑrhΔT)-σ2(t)2τmu]p(u,t)=Q(u,t)p(u,t),J(u,t)=\,\left[{-u+R\,I(t)\over\tau_{m}}+{\Delta_{T}\over\tau_{m}}\exp\left({u% -\vartheta_{rh}\over\Delta_{T}}\right)-{\sigma^{2}(t)\over 2\tau_{m}}{\partial% \over\partial u}\right]\,p(u,t)=Q(u,t)\,p(u,t)\,, (13.49)

where we have introduced on the right-hand side a linear operator QQ that comprises all the terms inside the square brackets. The stationary state, that we have analyzed in the previous subsection under the assumption of constant input I(t)=I0I(t)=I_{0} has a flux J0(u)=Q0(u)p0(u)J_{0}(u)=Q_{0}(u)\,p_{0}(u). In the presence of the periodic perturbation, the flux is

J(u,t)=J0(u)+J1(u)cos(ωt+ϕJ(u))J(u,t)=J_{0}(u)+J_{1}(u)\cos(\omega t+\phi_{J}(u)) (13.50)


J1(u)cos(ωt+ϕJ(u))=Q0(u)p1(u)cos(ωt+ϕp(u))+Q1(u,t)p0(u)+0order(ϵ2),J_{1}(u)\cos(\omega t+\phi_{J}(u))=Q_{0}(u)\,p_{1}(u)\cos(\omega t+\phi_{p}(u)% )+Q_{1}(u,t)\,p_{0}(u)+0_{\rm order}(\epsilon^{2}), (13.51)

where Q1(u,t)=ϵRcos(ωt)/τmQ_{1}(u,t)=\epsilon\,R\,\cos(\omega\,t)/\tau_{m} is the change of the operator QQ to order ϵ\epsilon. Note that the flux through the threshold θreset\theta_{\rm reset} gives the periodic modulation of the population activity.

Fig. 13.9: Frequency response of a population of exponential integrate-and-fire neurons, with parameter settings as in Fig. 13.8. Solid lines: Theory. Dashed line: analytical prediction for high frequencies. A. Case (i): The population response A1(ω)A_{1}(\omega) (top) to a periodic stimulation of frequency ω=2π/T\omega=2\pi/T in the situation of low noise (σ=2\sigma=2mV) shows a resonance at the population firing rate A0=44A_{0}=44Hz. The phase shift ϕ(ω)\phi(\omega) for low noise (bottom) approaches -45o-45^{o} for high frequencies. Case (ii): The population response A1(ω)A_{1}(\omega) for high noise (σ=6\sigma=6mV, population firing rate A0=5.6A_{0}=5.6Hz) has no resonance (top) and the phase shift ϕ(ω)\phi(\omega) approaches -90o-90^{o} (bottom). The combination of amplitude and phase A1(ω)exp(iϕ(ω))A_{1}(\omega)\exp(i\phi(\omega)) defines the complex frequency-dependent gain factor G^(ω)\hat{G}(\omega) in response to modulation of the input current. B. Gain factor GσG_{\sigma} for a modulation of the noise strength σ2(t)\sigma^{2}(t) while the input current is constant. Case (i): low noise; case (ii): high noise. A and B adapted from Richardson (434).

We insert our variables A,p,JA,p,J into the differential equation (13.38) and find that the stationary terms with subscript zero cancel each other, as it should be, and only the terms with subscript ‘1’ survive. To simplify the notation, it is convenient to switch from real to complex numbers and include the phase into the definition of A1,p1(u),J1(u)A_{1},p_{1}(u),J_{1}(u), e.g. A^1=A1exp(iϕA)\hat{A}_{1}=A_{1}\,\exp(i\phi_{A}); the hat indicates the complex number. If we take the Fourier transform over time, Eq. (13.38) becomes

-uJ^1(u)=iωp^1(u)+A^1[δ(u-θreset)-δ(u-ur)].\qquad-{\partial\over\partial u}\hat{J}_{1}(u)=i\omega\hat{p}_{1}(u)+\hat{A}_{% 1}\,[\delta(u-\theta_{\rm reset})-\delta(u-u_{r})]\,. (13.52)

Before we proceed further, let us have a closer look at Eqs. (13.51) and (13.52). We highlight three aspects (434). The first observation is that we have, quite arbitrarily, normalized the membrane potential density to an integral of unity. We could have chosen, with equal rights, a normalization to the total number NN of neurons in the population. Then the flux on the left-hand side of Eq. (13.52) would be enlarged by a factor of NN, but so would the membrane potential density and the population activity, which appear on the right-hand side. We could also multiply both sides of the equation by any other (complex) number. As a consequence, we can, for example, try to solve Eq. (13.52) for a population activity modulation A^1=1\hat{A}_{1}=1 and take care of the correct normalization only later.

The second observation is that the flux J1J_{1} in Eq. (13.51) can be quite naturally separated into two components. The first contribution to the flux is proportional to the perturbation p1p_{1} of the membrane potential density. The second component is caused by the direct action of the external current Q1(u,t)=ϵRcos(ωt)/τmQ_{1}(u,t)=\epsilon\,R\,\cos(\omega\,t)/\tau_{m}. We will exploit this separability of the flux in the following.

The third observation is that, for the case of exponential integrate-and-fire neurons, the explicit expression for the operators Q0Q_{0} and Q1Q_{1} in can be inserted into Eq. (13.51) which yields a first-order differential equation

up^1(u)=2σ2[-u+RI0+ΔTexp(u-ϑrhΔT)]p^1(u)+2Rϵσ2p0(u)-2τσ2J^1(u).{\partial\over\partial u}\hat{p}_{1}(u)={2\over\sigma^{2}}\,\left[-u+R\,I_{0}+% \Delta_{T}\exp\left({u-\vartheta_{rh}\over\Delta_{T}}\right)\right]\hat{p}_{1}% (u)+{2R\,\epsilon\over\sigma^{2}}\,p_{0}(u)-{2\tau\over\sigma^{2}}\,\hat{J}_{1% }(u). (13.53)

We therefore have two first-order differential equations (13.52) and (13.53) which are coupled to each other. In order to solve the two equations, we now exploit our second observation and split the flux into two components. We drop the hats on J1,p1,A1J_{1},p_{1},A_{1} to lighten the notation.

(i) A ‘free’ component J1free(u)J_{1}^{\rm free}(u) describes the flux that would occur in the absence of external drive (ϵ=0\epsilon=0), but in the presence of a periodically modulated population activity A1A_{1}. Intuitively, the distribution of the membrane potential density p1free(u)p_{1}^{\rm free}(u) and the flux J1freeJ_{1}^{\rm free} must exhibit some periodic ‘breathing pattern’ to enable the periodic modulation of the flux through the threshold of strength A^1\hat{A}_{1}. We find the ‘free’ component by integrating Eq. (13.53) with ϵ=0\epsilon=0 in parallel with Eq. (13.52) with parameter A1=1A_{1}=1 (i.e. the periodic flow through threshold is of unit strength). The integration starts at the initial condition p1(θreset)=0p_{1}(\theta_{\rm reset})=0 and J1free(θreset)=A1=1J_{1}^{\rm free}(\theta_{\rm reset})=A_{1}=1 and continues toward decreasing voltage values. Integration stops at a lower bound ulowu_{\rm low} which we place at an arbitrarily large negative value.

(ii) A ‘driven’ component J1ϵJ_{1}^{\rm\epsilon} accounts for the fact that the periodic modulation is in fact caused by the input. We impose for the ‘driven’ component that the modulation of the flux through threshold vanishes: A1=0A_{1}=0. We can therefore find the ‘driven’ component by integrating Eq. (13.53) with a finite ϵ>0\epsilon>0 in parallel with Eq. (13.52) with parameter A1=0A_{1}=0 starting at the initial condition p1(θreset)=0p_{1}(\theta_{\rm reset})=0 and J1ϵ(θreset)=A1=0J_{1}^{\rm\epsilon}(\theta_{\rm reset})=A_{1}=0 and integrating toward decreasing voltage values. As before, integration stops at a lower bound ulowu_{\rm low} which we can shift to arbitrarily large negative values.

Finally we add the two solutions together: Any combination of parameters a1,a2a_{1},a_{2} for the total flux a1J1free(u)+a2J1ϵ(u)a_{1}\,J_{1}^{\rm free}(u)+a_{2}\,J_{1}^{\epsilon}(u) and the total density a1p1free(u)+a2p1ϵ(u)a_{1}\,p_{1}^{\rm free}(u)+a_{2}\,p_{1}^{\epsilon}(u) will solve the pair of differential equations (13.53) and (13.52). Which one is the right combination?

The total flux must vanish at large negative values. Therefore we require a boundary condition

0=J(ulow)=a1J1free(ulow)+a2J1ϵ(ulow).0=J(u_{\rm low})=a_{1}\,J_{1}^{\rm free}(u_{\rm low})+a_{2}\,J_{1}^{\epsilon}(% u_{\rm low})\,. (13.54)

We recall that J1ϵ(u)J_{1}^{\epsilon}(u) is proportional to the drive ϵ\epsilon. Furthermore, the initial condition was chosen such that J1free(θreset)J_{1}^{\rm free}(\theta_{\rm reset}) yields a population response A1=1A_{1}=1 so that, in the combined solution, the factor a1a_{1} is the population response!

We are interested in the ’gain factor’ at stimulation frequency ω\omega. We started this section by applying a periodic stimulus of strength ϵ\epsilon and observing, at the same periodic frequency a modulation of strength A1A_{1}. The gain factor is defined as G^(ω)=A1/ϵ\hat{G}(\omega)=A_{1}/\epsilon. With the above arguments the gain factor is (434)

G^(ω)=a1ϵ=-a2ϵJ1ϵ(ulow)J1free(ulow).\hat{G}(\omega)={a_{1}\over\epsilon}=-{a_{2}\over\epsilon}\,{J_{1}^{\epsilon}(% u_{\rm low})\over J_{1}^{\rm free}(u_{\rm low})}\,. (13.55)

The gain factor G^(ω)\hat{G}(\omega) has an amplitude |G^(ω)||\hat{G}(\omega)| and a phase ϕG\phi_{G}. Amplitude and phase of the gain factor of a population of exponential integrate-and-fire neurons are plotted in Fig. 13.9A.

The same numerical integration scheme that starts at the numerical threshold θreset\theta_{\rm reset} and integrates downward with appropriate initial conditions can also be applied to other, linear as well as nonlinear, neuron models. Furthermore, with the same scheme it is also possible to calculate the gain factor GσG_{\sigma} in response to a modulation of the variance σ2(t)\sigma^{2}(t) (Fig. 13.9B), or the response GϑG_{\vartheta} to periodic modulation of model parameters such as the rheobase threshold ϑrh\vartheta_{rh} (434). For earlier results on GσG_{\sigma} see also (78; 298; 477).