To understand the functional implications of the variable onset potential, I constructed a phenomeno-logical neuron model, the Gaussian neuron model with subthreshold adaptation (V−ψ model). In this model, unlike in other phenomenological models, such as the LIF model, the dynamics of the MP is not defined by a differential equation, but directly as a Gaussian stochastic process V(t)with given statistical properties. This process models the fluctuating MP, which is typical for cortical neurons in vivo (see Fig. 4.3). The theory of Gaussian random fields was first introduced into theoretical neu-roscience in (Wolf & Geisel,1998), to describe the dynamics of pinwheels in orientation selectivity maps in the visual cortex.
The statistical properties of the MP fluctuations are completely characterized by the correlation func-tion of the MP C(τ)and its mean valuehV(t)i, assuming that the fluctuations of the subthreshold MP are stationary. In the following analysis I will assume that the correlation function is of the form:
C(τ) =A(k2exp(−k1|t|)−k1exp(−k2|t|)) (5.2) This choice ensures that the correlation function is twice differentiable at t=0, thus ensuring that the variance of the first temporal derivative of the MP exists. By choosing the parameters A=√
2σV3/σV˙,
where the dots denote first temporal derivatives.
So far, the model includes only passive membrane properties. AP generation is incorporated into the model by assuming a threshold potentialψ(t), which explicitely depends on time. Each time the MP V(t) crosses the time-dependent threshold from below, an AP is emitted. An important difference to the classical IF model is that there is no reset mechanism incorporated into the model. This may appear as a very crude simplification as the occurrence of an AP in a real neuron dramatically changes the MP time course and the intrinsic states of all voltage-gated channels. To reassure that we don’t make a large error by introducing this approximation, it is expedient to first define the dynamical regime in which I will use the model in the following.
The correlation time of the MP V(t)is given by:
τc= 3.4 ms. The average firing rate on the other hand does hardly ever exceed 10 Hz, the spontaneous firing rate of a cortical neuron in vivo is, typically, even much lower<5 Hz. The mean ISI is then at least 100 ms. This means that if the firing rate is reasonably small, the MP will not “remember” if it was reset after an AP was generated. The big advantage of the applied approximation is that many stationary and dynamic response properties can be calculated without any approximations. This is not
5.4. The Gaussian neuron model with subthreshold adaptation (V−ψ model)
Figure 5.2: Sample trajectory of theV−ψ model. The subthreshold MP trajectory is shown in blue, the time-dependent threshold in red. Whenever the MP reaches the threshold, an AP is generated (indicated as blue vertical lines). To simulate the Gaussian process, a two-dimensional Ornstein-Uhlenbeck process was used. Parameters:σV=2 mV,σV˙ =1 mV/ms,ψ0=5 mV,τψ=30 ms.
possible in standard IF models with realistic input currents, as I discussed in Chap.3.
The dynamics of the threshold is modeled by a first order kinetics, driven by the MP V(t)with a time constantτψ:
τψψ(t) = (ψ˙ 0−ψ(t)) +c(V(t)−V0), (5.6) where ψ0 is the mean threshold voltage and c the coupling between the threshold and the MP. In Fig.5.2, a sample realization of a model trajectory is displayed. Whenever the MP crosses the time-dependent threshold from below, an AP is emitted.
As I am interested in the population coding of the V−ψ model, in the following population aver-aged quantities will be considered. As in the assessment of the population coding properties of the generalized θ-neuron (cf. chapter5), it will be assumed that the MP fluctuations in each neuron are independent. Before discussing the impact of the time varying threshold on the dynamic response properties, it will be shown, how stationary response properties are calculated in this model.
The average number of APs in a population of neurons in the time interval(t,t+∆t]is given by:
hNi =
where the angular brackets h·i denote population averages. In the last equation, I introduced the time-dependent firing rateν(t), which is defined as:
ν(t) = Z ∞
−∞dV(t)d ˙V(t)dψ(t)δ(V(t)−ψ(t))
V˙(t)−ψ˙(t)
Θ(V˙(t))P(V(t),V˙(t),ψ(t)) (5.10)
Hereδ(·)denotes the Dirac distribution andΘ(·)is the Heaviside function. The joint pdf of V(t), ˙V(t)
For the case of a constant threshold,ψ=ψ0, the firing rate reduces to:
νFT= (2π σvσV˙)−1 since for a stationary process V(t), the covariance
V(t)V˙(t)
is zero. The integral for the firing rate can be solved explicitly in this case, yielding:
νFT=√
With threshold adaptation, the correlation matrix C also includes the covariances between the thresh-oldψ(t), the MP V(t)and its velocity ˙V(t):
In this case, the rate is given by the double integral:
ν = τψ−1
in which the threshold dynamics is included implicitly in the pdf P(·,·,·)and in the modulus in the integral kernel. In Fig.5.3, the stationary firing rate as a function of τψ for various values of c is displayed. For values of c close to 1, the rate is almost zero forτψ→0 and increases to the valueνFT
predicted by Eq. (5.13). For decreasing values of c, the rate increases but always stays belowνFT. Intuitively, for small values ofτψ the threshold follows the MP, thus suppressing all threshold cross-ings. Forτψ→∞, the threshold becomes constant and its varianceσψ2 goes to zero as 3c2σV3/√
2σV˙τψ
. Thus, in this limit the model approaches the stationary rate of the Gaussian neuron with a fixed
thresh-5.4. The Gaussian neuron model with subthreshold adaptation (V−ψ model)
0 10 20 30 40 50
0 1 2
Threshold time constant τψ (ms)
Stationary rate ν (Hz)
Figure 5.3: Stationary rate of theV−ψmodel. Solid lines: Stationary rateνcalculated via Eq. (5.16) for different values of the threshold coupling constantc(0.01,0.1,0.25,0.5,0.75,1.0). Dashed line:
Rate in the limitc→0.
The rate increases for larger values of τψ, as the threshold integrates over an increasingly long subthreshold MP trajectory. When the coupling between the threshold and then MP is increased, the rate decreases. Parameters:σV=2 mV,σV˙=3 mV/ms,ψ0=6 mV.
old.