Subsections

• 4.3.1 Reduction of the Hodgkin-Huxley Model
  • 4.3.1.1 The $\eta$-kernel
  • 4.3.1.2 The $\kappa$-kernel
  • 4.3.1.3 The threshold $\vartheta$
  • 4.3.1.4 Input scenarios
  • 4.3.1.5 Example: Stimulation by time-dependent input
  • 4.3.1.6 Example: Constant input and mean firing rates
  • 4.3.1.7 Example: Step current input
  • 4.3.1.8 Example: Spike input
  
  
• 4.3.2 Reduction of a Cortical Neuron Model
  • 4.3.2.1 Reduction to a nonlinear integrate-and-fire model
  • 4.3.2.2 Example: Constant input
  • 4.3.2.3 Example: Fluctuating input
  • 4.3.2.4 Reduction to a Spike Response Model
  • 4.3.2.5 Example: Fluctuating input
  
  
• 4.3.3 Limitations

4.3 From Detailed Models to Formal Spiking Neurons

In this section we study the relation between detailed conductance-based neuron models and formal spiking neurons as introduced above. In Section 4.3.1, we discuss how an approximative mapping between the Spike Response Model and the Hodgkin-Huxley model can be established. While the Hodgkin-Huxley model is of type II, cortical neurons are usually described by type I models. In Subsection 4.3.2 we focus on a type-I model of cortical interneurons and reduce it systematically to different variants of spiking neuron models, in particular to a nonlinear integrate-and-fire model and a Spike Response Model. In all sections, the performance of the reduced models is compared to that of the full model. To do so we test the models with a constant or fluctuating input current.

4.3.1 Reduction of the Hodgkin-Huxley Model

The system of equations proposed by Hodgkin and Huxley (see Chapter 2.2) is rather complicated. It consists of four coupled nonlinear differential equations and as such is difficult to analyze mathematically. For this reason, several simplifications of the Hodgkin-Huxley equations have been proposed. The most common approach reduces the set of four differential equations to a two-dimensional problem as discussed in Chapter 3. In this section, we will take a somewhat different approach to reduce the four Hodgkin-Huxley equations to a single variable u(t), the membrane potential of the neuron (Kistler et al., 1997). As we have seen in Fig. 2.4B, the Hodgkin-Huxley model shows a sharp, threshold-like transition between an action potential (spike) for a strong stimulus and a graded response (no spike) for a slightly weaker stimulus. This suggests the idea that emission of an action potential can be described by a threshold process. We therefore aim for a reduction towards a spiking neuron model where spikes are triggered by a voltage threshold. Specifically, we will establish an approximative mapping between the Spike Response Model and the Hodgkin-Huxley model.

Action potentials in the Hodgkin-Huxley model have the stereotyped time course shown in Fig. 2.4A. Whatever the stimulating current that has triggered the spike, the form of the action potential is always roughly the same (as long as the current stays in a biologically realistic regime). This is the major observation that we will exploit in the following. Let us consider the spike that has been triggered at time $\hat{{t}}$. If no further input is applied for t > $\hat{{t}}$, the voltage trajectory will have a pulse-like excursion before it eventually returns to the resting potential. For t > $\hat{{t}}$, we may therefore set u(t) = $\eta$(t - $\hat{{t}}$) + u_rest where $\eta$ is the standard shape of the pulse and u_rest is the resting potential. We have $\eta$(t - $\hat{{t}}$)$\to$ 0 for t - $\hat{{t}}$$\to$$\infty$, because, without further input, the voltage will eventually approach the resting value.

Let us now consider an additional small input current pulse I which is applied at t > $\hat{{t}}$. Due to the input, the membrane potential will be slightly perturbed from its trajectory. If the input current is sufficiently small, the perturbation can be described by a linear impulse response function $\kappa$. The response to an input pulse, and therewith the response kernel $\kappa$, can depend on the arrival time of the input relative to the last spike at $\hat{{t}}$. For an input with arbitrary time course I(t) we therefore set

$$\eta \hat{{t}} \int_{0}^{{t-\hat{t}}} \kappa \hat{{t}}$$ (4.63)

Equation (4.63) is a special case of the Spike Response Model (SRM) introduced in Chapter 4.2. Note that after an appropriate shift of the voltage scale the resting potential can always be set to zero, u_rest = 0.

To construct an approximative mapping between the SRM (4.63) and the Hodgkin-Huxley equations, we have to determine the following three terms: (i) the kernel $\eta$ which describes the response to spike emission, (ii) the kernel $\kappa$ which describes the response to incoming current, and (iii) the value of the threshold $\vartheta$.

4.3.1.1 The $\eta$-kernel

In the absence of input the membrane potential u is at its resting value u_rest. If we apply a strong current pulse, an action potential will be triggered. The time course of the action potential determines the kernel $\eta$.

To find the kernel $\eta$ we use the following procedure. We take a square current pulse of the form

$${q_0\over \Delta} \Theta \Theta \Delta$$ (4.64)

with duration $\Delta$ = 1ms, a unit charge q₀, and c a parameter chosen large enough to evoke a spike. The kernel $\eta$ allows us to describe the standard form of the spike and the spike after-potential. We set

$$\eta \hat{{t}} \Theta \hat{{t}}$$ (4.65)

Here, u(t) is the voltage trajectory caused by the supra-threshold current pulse. The firing time $\hat{{t}}$ is defined by the moment when u crosses the threshold $\vartheta$ from below. The kernel $\eta$(s) with its pronounced hyperpolarizing spike after-potential that extends over more than 15ms is shown in Fig. 4.12A.

Figure 4.12: A. The action potential of the Hodgkin-Huxley model defines the kernel $\eta$. The spike has been triggered at t = 0. B. The voltage response of the Hodgkin-Huxley model to a short sub-threshold current pulse defines the kernel $\kappa$. The input pulse has been applied at t = 0. The last output spike occurred at $\hat{{t}}$ = - $\Delta$t. We plot the time course $\kappa$($\Delta$t + t, t). For $\Delta$t$\to$$\infty$ we get the response shown by the solid line. For finite $\Delta$t (dashed line, output spike $\Delta$t = 10.5 ms before the input spike; dotted line $\Delta$t = 6.5 ms), the duration of the response is reduced due to refractoriness; cf. Fig. 2.7B. Taken from [Kistler et al., 1997].

4.3.1.2 The $\kappa$-kernel

The kernel $\kappa$ characterizes the linear response of the neuron to a weak input current pulse. To measure $\kappa$ we use a first strong pulse to initiate a spike at a time $\hat{{t}}$ < 0 and then apply a second weak pulse at t = 0. The second pulse is a short stimulus as in Eq. (4.64), but with a small amplitude so that nonlinear effects in the response can be neglected. The result is a membrane potential with time course u(t). Without the second pulse the time course of the potential would be u₀(t) = $\eta$(t - $\hat{{t}}$) + u_rest for t > $\hat{{t}}$. The net effect of the second pulse is u(t) - u₀(t), hence

$$\kappa \hat{{t}} {1\over c} \left[\vphantom{u(t) - \eta(t-\hat{t}) -u_{\rm rest} }\right. \eta \hat{{t}} \left.\vphantom{u(t) - \eta(t-\hat{t}) -u_{\rm rest} }\right]$$ (4.66)

We repeat the above procedure for various spike times $\hat{{t}}$.

The result is shown in Fig. 4.12. Since the input current pulse delivers its charge during a very short amount of time, the $\kappa$-kernel jumps instantaneously at time t = 0 to a value of 1mV. Afterwards it decays, with a slight oscillation, back to zero. The oscillatory behavior is characteristic for type II neuron models (Izhikevich, 2001). The decay of the oscillation is faster if there has been a spike in the recent past. This is easy to understand intuitively. During and immediately after an action potential many ion channels are open. The resistance of the cell membrane is therefore reduced and the effective membrane time constant is shorter; cf. Fig. 2.7B.

4.3.1.3 The threshold $\vartheta$

The third term to be determined is the threshold $\vartheta$ which we will take as fixed. Even though Fig. 2.4B suggests that the Hodgkin-Huxley equations exhibit a certain form of threshold behavior, the threshold is not well-defined (Koch et al., 1995; Rinzel and Ermentrout, 1989) and it is fairly difficult to estimate a voltage threshold directly from a single series of simulations. We therefore take the threshold as a free parameter which will be adjusted by a procedure discussed below.

4.3.1.4 Input scenarios

In order to test the fidelity of the Spike Response Model we use the same input scenarios as in Chapter 2.2 for the Hodgkin-Huxley model. In particular, we consider constant input current, step current, and flucutating input current. We start with the time-dependent fluctuating input, since this is probably the most realistic scenario. We will see that the Spike Response Model with the kernels that have been derived above can approximate the spike train of the Hodgkin-Huxley model to a high degree of accuracy.

4.3.1.5 Example: Stimulation by time-dependent input

To test the quality of the SRM approximation we compare the spike trains generated by the Spike Response Model (4.63) with that of the full Hodgkin-Huxley model (2.4)-(2.6). We study the case of a time-dependent input current I(t) generated by the procedure discussed in section 2.2.2; cf. Fig. 2.7. The same current is applied to both the Hodgkin-Huxley and the Spike Response model. The threshold $\vartheta$ of the Spike Response Model has been adjusted so that the total number of spikes was about the same as in the Hodgkin-Huxley model; see Kistler et al. (1997) for details. In Fig. 4.13 the voltage trace of the Hodgkin-Huxley model is compared to that of the Spike Response Model with the kernels $\eta$ and $\kappa$ derived above. We see that the approximation is excellent both in the absence of spikes and during spiking. As an aside we note that it is indeed important to include the dependence of the kernel $\kappa$ upon the last output spike time $\hat{{t}}$. If we neglected that dependence and used $\kappa$($\infty$, s) instead of $\kappa$(t - $\hat{{t}}$, s), then the approximation during and immediately after a spike would be significantly worse; see the dotted line in the lower right graph of Fig. 4.13.

Figure 4.13: A segment of the spike train of Fig. 2.7. The inset in the lower left corner shows the voltage of the Hodgkin-Huxley model (solid) together with the approximation of the Spike Response Model defined by (4.63) (long-dashed line) during a period where no spike occurs. The approximation is excellent. The inset on the lower right shows the situation during and after a spike. Again the approximation by the long-dashed line is excellent. For comparison, we also show the approximation by the SRM₀ model which is significantly worse (dotted line). Taken from Kistler et al. (1997).

To check whether both models generated spikes at the same time we introduce the coincidence rate

$$\Gamma {N_{\rm coinc} - \langle N_{\rm coinc}\rangle \over {1\over 2}(N_{\rm SRM} + N_{\rm full})} {1\over \mathcal{N}}$$ (4.67)

where N_SRM is the number of spikes of the Spike Response Model, N_full is the number of spikes of the full Hodgkin-Huxley model, N_coinc is the number of coincidences with precision $\Delta$, and $\langle$N_coinc$\rangle$ = 2 $\nu$ $\Delta$ N_full is the expected number of coincidences generated by a homogeneous Poisson process with the same rate $\nu$ as the Spike Response Model. The factor $\mathcal {N}$ = 1 - 2 $\nu$ $\Delta$ normalizes $\Gamma$ to a maximum value of one which is reached if the spike train of the Spike Response Model reproduces exactly that of the full model. A homogeneous Poisson process with the same number of spikes as the Spike Response Model would yield $\Gamma$ = 0.

We find that the Spike Response Model (4.63) reproduces the firing times and the voltage time course of the Hodgkin-Huxley model to a high degree of accuracy; cf. Fig. 4.13. More precisely, the Spike Response Model achieves with a fluctuating input current a coincidence rate $\Gamma$ of about 0.85 (Kistler et al., 1997). On the other hand, a leaky integrate-and-fire model with optimized time constant and fixed threshold yields coincidence rates in the range of only 0.45. The difference in the performance of Spike Response and integrate-and-fire model is not too surprising because the Spike Response Model accounts for the hyperpolarizing spike after-potential; cf. Fig 4.12A. In fact, an integrate-and-fire model with spike after-potential (or equivalently a dynamic threshold) achieves coincidence rates in the range of $\Gamma$ $\approx$ 0.7 (Kistler et al., 1997). Furthermore, the $\kappa$-kernel of the Spike Response Model describes the reduced responsiveness of the Hodgkin-Huxley model immediately after a spike; cf. Fig 4.12B. The model SRM₀ (with a kernel $\kappa$ that does not depend on t - $\hat{{t}}$) yields a coincidence rate $\Gamma$ that is significantly lower than that of the full Spike Response Model.

4.3.1.6 Example: Constant input and mean firing rates

We study the response of the Spike Response Model to constant stimulation using the kernels derived by the procedure described above. The result is shown in Fig. 4.14. As mentioned above, we take the threshold $\vartheta$ as a free parameter. If $\vartheta$ is optimized for stationary input, the frequency plots of the Hodgkin-Huxley model and the Spike Response Model are rather similar. On the other hand, if we took the value of the threshold that was found for time-dependent input, the current threshold for the Spike Response Model would be quite different as shown by the dashed line in Fig. 4.14.

Figure 4.14: The firing rate $\nu$ of the Hodgkin-Huxley model (solid line) is compared to that of the Spike Response Model. Two cases are shown. If the threshold $\vartheta$ is optimized for the constant-input scenario, we get the dotted line. If we take the same value of the threshold as in the dynamic-input scenario of the previous figure, we find the long-dashed line. Input current has a constant value I₀. Taken from Kistler et al. (1997).

4.3.1.7 Example: Step current input

As a third input paradigm, we test the Spike Response Model with step current input. For $\vartheta$ we take the value found for the scenario with time-dependent input. The result is shown in Fig. 4.15. The Spike Response Model shows the same three regimes as the Hodgkin-Huxley model. In particular, the effect of inhibitory rebound is present in the Spike Response Model. The location of the phase boundaries depends on the choice of $\vartheta$.

Figure 4.15: Phase diagram of the Spike Response Model for stimulation with a step current. Kernels $\epsilon$ and $\eta$ are adapted to the Hodgkin-Huxley model. The current I is switched at t = 0 from I₁ to I₂. The y-axis is the step size $\Delta$I. Three regimes denoted by S, R, and I may be distinguished. In I no action potential is initiated (inactive regime). In S, a single spike is initiated by the current step (single spike regime). In R, periodic spike trains are triggered by the current step (repetitive firing). Examples of voltage traces in the different regimes are presented in the smaller graphs to the left and right of the phase diagram in the center. The phase diagram should be compared to that of the Hodgkin-Huxley model in Fig. 2.6. Taken from (Kistler et al., 1997).

4.3.1.8 Example: Spike input

In the Hodgkin-Huxley model (2.4), input is formulated as an explicit driving current I(t). In networks of neurons, input typically consists of the spikes of presynaptic neurons. Let us, for the sake of simplicity, assume that a spike of a presynaptic neuron j which was emitted at time t_j^(f) generates in the postsynaptic neuron i a current I(t) = w_ij $\alpha$(t - t_j^(f)). Here, $\alpha$ describes the time course of the postsynaptic current and w_ij scales the amplitude of the current. The voltage of the postsynaptic neuron i changes, according to (4.63) by an amount $\Delta$u_i(t) = w_ij $\int_{0}^{{t-\hat{t}_i}}$$\kappa$(t - $\hat{{t}}_{i}^{}$, s) $\alpha$(t - t_j^(f) - s) ds, where $\hat{{t}}_{i}^{}$ is the last output spike of neuron i. The voltage response $\Delta$u_i to an input current of unit amplitude (w_ij = 1) defines the postsynaptic potential $\epsilon$, hence

$$\epsilon \hat{{t}}_{i}^{} \int_{0}^{{t-\hat{t}_i}} \kappa \hat{{t}}_{i}^{} \alpha$$ (4.68)

What is the meaning of the definition (4.68)? If several presynaptic neurons j transmit spikes to neuron i, then the total membrane potential of the postsynaptic neuron is in analogy to Eq. (4.63)

$$\eta \hat{{t}}_{i}^{} \sum_{j}^{} \sum_{f}^{} \epsilon \hat{{t}}_{i}^{}$$ (4.69)

Equation (4.69) is the standard equation of the Spike Response Model. We emphasize that the time course of the postsynaptic potential depends on t - $\hat{{t}}_{i}^{}$; the first argument of $\epsilon$ takes care of this dependence.

4.3.2 Reduction of a Cortical Neuron Model

We have seen in the previous section that the Spike Response Model can provide a good quantitative approximation of the Hodgkin-Huxley model. Though the Hodgkin-Huxley equation captures the essence of spike generation it is ``only'' a model of the giant axon of the squid which has electrical properties that are quite different from those of cortical neurons we are mostly interested in. The natural question is thus whether the Spike Response Model can also be used as a quantitative model of cortical neurons. In the following we discuss a conductance-based neuron model for a cortical interneuron and show how such a model can be reduced to a (nonlinear) integrate-and-fire model or to a Spike Response Model.

The starting point is a conductance-based model that has originally been proposed as a model of fast-spiking neo-cortical interneurons (Erisir et al., 1999). We have chosen this specific model for two reasons: First, just as most other cortical neuron models, this model has - after a minor modification - a continuous gain function (Lewis and Gerstner, 2001) and can hence be classified as a type I model (Ermentrout, 1996). This is in contrast to the Hodgkin-Huxley model which exhibits a discontinuity in the gain function and is hence type II. Second, this is a model for interneurons that show little adaptation, so that we avoid most of the complications caused by slow ionic processes that cannot be captured by the class of spiking neuron models reviewed above. Furthermore, the model is comparatively simple, so that we can hope to illustrate the steps necessary for a reduction to formal spiking neuron models in a transparent manner.

The model neuron consists of a single compartment with a non-specific leak current and three types of ion current, i.e., a Hodgkin-Huxley type sodium current I_Na = g_Na m³ h (u - E_Na), a slow potassium current I_slow = g_Kslow n⁴_slow (u - E_K), and a fast potassium current I_fast = g_Kfast n²_fast (u - E_K). The response properties of the cortical neuron model to pulse input and constant current have already been discussed in Chapter 2.3; cf. Fig. 2.11. We now want to reduce the model to a nonlinear integrate-and fire model or, alternatively, to a Spike Response Model. We start with the reduction to an integrate-and-fire model.

Table 4.1: Cortical neuron model. The equilibrium value x₀(u) = $\alpha$/($\alpha$ + $\beta$) is reached with a time constant $\tau_{x}^{}$(u) = 1/($\alpha$ + $\beta$) where x stands for one of the gating variables m, h, n_slow, n_fast. Membrane capacity C = 1.0 $\mu$F/cm².

4.3.2.1 Reduction to a nonlinear integrate-and-fire model

In order to reduce the dynamics of the full cortical neuron model to that of an integrate-and-fire model we proceed in two steps. As a first step, we keep all variables, but introduce a threshold for spike initiation. We call this the multi-current integrate-and-fire model. In the second step, we separate gating variables into fast and slow ones. `Fast' variables are replaced by their steady state values, while `slow' variables are replaced by constants. The result is the desired nonlinear integrate-and-fire model with a single dynamical variable.

In step (i), we make use of the observation that the shape of an action potential of the cortical neuron model is always roughly the same, independently of the way the spike is initiated; cf. Fig. 4.16. Instead of calculating the shape of an action potential again and again, we can therefore simply stop the costly numerical integration of the nonlinear differential equations as soon as a spike is triggered and restart the integration after the down-stroke of the spike about 1.5-2ms later. We call such a scheme a multi-current integrate-and-fire model. The interval between the spike trigger time $\hat{{t}}$ and the restart of the integration corresponds to an absolute refractory period $\Delta^{{\rm abs}}_{}$.

Figure 4.16: Dynamics during a spike. A. Action potentials have roughly the same shape, whether they are triggered by a constant current of 5$\mu$A/cm² (solid line) or by a single 2-ms-current pulse of amplitude 20 $\mu$A/cm² (dashed line). B. The time course of the gating variable h is, however, significantly different. The time s = 0 markes the moment when the voltage rises above -40 mV. The vertical dotted line indicates the absolute refractory period $\Delta^{{\rm abs}}_{}$ =1.7ms.

In order to transform the cortical neuron model into a multi-current integrate-and-fire model we have to define a voltage threshold $\vartheta$, a refractory time $\Delta^{{\rm abs}}_{}$, and the reset values from which the integration is restarted. We fix the threshold at $\vartheta$ = - 40 mV; the exact value is not critical and we could take values of -20mV or -45mV without changing the results. For $\vartheta$ = - 40 mV, a refractory time $\Delta^{{\rm abs}}_{}$ = 1.7ms and a reset voltage u_r = - 85 mV is suitable; cf. Fig. 4.16.

To restart the integration of the differential equation we also have to specify initial conditions for the gating variables m, h, n_slow, and n_fast. This, however, involves a severe simplification, because their time course is not as stereotyped as that of the membrane potential, but depends on the choice of the input scenario; cf. Fig. 4.16B. In the following we optimize the reset values for a scenario with a constant input current I^ext = 5 $\mu$A/cm² that leads to repetitive firing at about 40Hz. The reset values are m_r = 0.0;h_r = 0.16;n_{slow, r} = 0.874;n_{fast, r} = 0.2; and u_r = - 85 mV. This set of parameters yields a near-perfect fit of the time course of the membrane potential during repetitive firing at 40Hz and approximates the gain function of the full cortical neuron model to a high degree of accuracy; cf. Fig. 4.17.

Figure 4.17: The gain function of the multi-current IF model A1 (dashed line) compared to that of the full model (solid line).

So far the model contains still all five variables u, m, h, n_slow, n_fast. To get rid of all the gating variables we distinguish between variables x that are either fast as compared to u - in which case we replace it by its steady-state value x₀(u) - or slow as compared to u - in which case we replace x by a constant. Here, m is the only fast variable and we replace m(t) by its steady-state value m₀[u(t)]. The treatment of the other gating variables deserves some extra discussion.

A thorough inspection of the time course of n_fast(t) shows that n_fast is most of the time close to its resting value, except for a 2ms interval during and immediately after the down-stroke of an action potential. If we take a refractory time of $\Delta^{{\rm abs}}_{}$ = 4 ms, most of the excursion trajectory of n_fast falls within the refractory period. Between spikes we can therefore replace n_fast by its equilibrium value at rest n_{fast, rest} = n_{0, fast}(u_rest).

The gating variables h and n_slow vary slowly, so that the variables may be replaced by averaged values h_av and n_{slow, av}. The average, however, depends on the input scenario. We stick to a regime with repetitive firing at 40Hz where h_av = 0.7 and n_{slow, av} = 0.8.

With m = m₀(u) and constant values for h, n_slow, n_fast, the dynamics of the full cortical neuron model reduces to

$${{\text{d}}u\over {\text{d}}t}$$ = g_Na [m₀(u)]³ h_av (u - E_Na) + g_Kslow n_{slow, av}² (u - E_K)

         + g_Kfast n_{fast, rest}⁴ (u - E_K) + g_l (u - E_l) + I^ext(t). (4.70)

After division by C, we arrive at a single nonlinear equation

$${{\text{d}}u\over {\text{d}}t} {1\over C}$$ (4.71)

The passive membrane time constant of the model is inversely proportional to the slope of F at rest: $\tau$ = | dF/du|^-1_u=urest. In principle the function F could be further approximated by a linear function with slope -1/$\tau$ and then combined with a threshold at, e.g., $\vartheta$ = - 45 mV. This would yield a linear integrate-and-fire model. Alternatively, F can be approximated by a quadratic function which leads us to a quadratic integrate-and-fire neuron; cf. Fig. 4.18.

Figure 4.18: The function F(u) of a nonlinear integrate-and-fire neuron (solid line) derived from a cortical neuron model is compared to a quadratic (dotted line) and a linear (long-dashed line) approximation. The linear approximation stops at the threshold $\vartheta$ = - 45 mV (vertical line).

To test the fidelity of the reduction to a nonlinear integrate-and-fire model, we compare its behavior to that of the full cortical neuron model for various input scenarios. It turns out that the behavior of the model is good as long as the mean firing rate is in the range of 40Hz, which is not too surprising given the optimization of the parameters for this firing rate. Outside the range of 40±10Hz there are substantial discrepancies between the reduced and the full model.

4.3.2.2 Example: Constant input

Let us focus on constant input first. With our set of parameters we get a fair approximation of the gain function, except that the threshold for repetitive firing is not reproduced correctly; cf. Fig. 4.19A. We note that the firing rate at a stimulation of 5$\mu$A/cm² is reproduced correctly which is no surprise since our choice of parameters has been based on this input amplitude.

4.3.2.3 Example: Fluctuating input

For a critical test of the nonlinear integrate-and-fire model, we use a fluctating input current with zero mean. The amplitude of the fluctuations determines the mean firing rate. The nonlinear integrate-and-fire model, however, does not reproduce the firing rate as a function of the fluctuation amplitude of the full model, except at $\nu$ $\approx$ 40Hz; cf. Figure 4.19B.

For a more detailed comparison of the nonlinear integrate-and-fire with the full model, we stimulate both models with the same fluctuating current. From Fig. 4.20A, we see that the voltage time course of the two models is most of the time indistinguishable. Occasionally, the nonlinear integrate-and-fire model misses a spike, or adds an extra spike. For this specific input scenario (where the mean firing rate is about 40Hz), a coincidence rate of about $\Gamma$ = 85 is achieved (based on a precision of $\Delta$ = ±2ms). Outside the regime of $\nu$ $\approx$ 40Hz, the coincidence rate $\Gamma$ breaks down drastically; cf. Fig. 4.20B.

Figure 4.19: A Gain function for stationary input. The firing rate vs. the input current of the full cortical neuron model (solid line) compareded to that of a nonlinear integrate-and-fire model (long-dashed line). B. Fluctuating input. The mean firing rate of the full cortical neuron model (solid line) compared to that of the nonlinear integrate-and-fire model (diamonds) as a function of the amplitude of the input fluctuations. At an amplitude of 0.4 (arbitrary units) both the full and the integrate-and-fire model fire at about 40Hz.

4.3.2.4 Reduction to a Spike Response Model

As a second approximation scheme, we consider the reduction of the conductance-based neuron model to a Spike Response Model. We thus have to determine the kernels $\eta$(t - $\hat{{t}}$), $\kappa$(t - $\hat{{t}}$, s), and adjust the (time-dependent) threshold $\vartheta$(t - $\hat{{t}}$). We proceed in three steps. As a first step we reduce the model to an integrate-and-fire model with spike-time dependent time constant. As a second step, we integrate the model so as to derive the kernels $\eta$ and $\kappa$. As a final step, we choose an appropriate spike-time dependent threshold.

Figure 4.20: A. The spike train of the nonlinear integrate-and-fire model (dashed line) compared to that of the full cortical neuron model (solid line). The integrate-and-fire model fires an extra spike at about t = 1652ms but misses the spike that occurs about 4ms later. Spikes are replaced by triangular pulses that span the refractory period of $\Delta^{{\rm abs}}_{}$ = 4ms. For this input scenario (viz. fluctuation amplitude 0.4), a coincidence rate of about 0.85 is achieved ( $\Delta$ = 2 ms). B. Comparison of the coincidence rates $\Gamma$ for the multi-current integrate-and-fire model (dotted line) and the nonlinear integrate-and-fire model (solid line). A value of $\Gamma$ = 1 implies perfectly coincident spike trains, while a value of $\Gamma$ = 0 implies that coincidences can be explained by chance. For the definition of $\Gamma$, see Eq. (4.67).

In step (i), we stimulate the full model by a short super-threshold current pulse in order to determine the time course of the action potential and its hyper-polarizing spike after-potential. Let us define $\hat{{t}}$ as the time when the membrane potential crosses an (arbitrarily fixed) threshold $\vartheta$, e.g., $\vartheta$ = - 50 mV. The time course of the membrane potential for t > $\hat{{t}}$, i.e., during and after the action potential, defines the kernel $\eta$(t - $\hat{{t}}$). If we were interested in a purely phenomenological model, we could simply record the numerical time course u(t) and define $\eta$(t - $\hat{{t}}$) = u(t) - u_rest for t > $\hat{{t}}$; cf. Section 4.3.1. It is, however, instructive to take a semi-analytical approach and study the four gating variables m, h, n_slow and n_fast. About 2ms after initiation of the spike, all four variables have passed their maximum or minimal values and are on their way back to equilibrium. We set $\Delta^{{\rm abs}}_{}$ = 2ms. For t$\ge$$\hat{{t}}$ + $\Delta^{{\rm abs}}_{}$, we fit the approach to equilibrium by an exponential

$$\left(\vphantom{ -{t-\hat{t}-\Delta^{\rm abs}\over \tau_x } }\right. {t-\hat{t}-\Delta^{\rm abs}\over \tau_x} \left.\vphantom{ -{t-\hat{t}-\Delta^{\rm abs}\over \tau_x } }\right)$$ (4.72)

where x = m, h, n_slow, n_fast stands for the four gating variables, $\tau_{x}^{}$ is a fixed time constant, x_r is the initial condition at t = $\hat{{t}}$ + $\Delta^{{\rm abs}}_{}$, and x_rest = x₀(u_rest) is the equilibrium value of the gating variable at the resting potential.

Given the time course of the gating variables, we know the conductance of each ion channel as a function of time. For example, the potassium current I_Kfast is

$$\hat{{t}}$$ (4.73)

where g_fast(t - $\hat{{t}}$) is an exponential function with time constant $\tau_{{n_{\rm fast}}}^{}$/2. We insert the time-dependent conductance into the current equation and find for t$\ge$$\hat{{t}}$ + $\Delta^{{\rm abs}}_{}$

$${{\text{d}}u\over {\text{d}}t} \sum_{j}^{} \hat{{t}}$$ (4.74)

Here, the sum runs over the four ion channels I_Na, I_Kslow, I_Kfast, and I_l. With the definition of an effective time constant, $\tau$(t - $\hat{{t}}$) = C/$\sum_{j}^{}$g_j(t - $\hat{{t}}$), and with I^ion = $\sum_{j}^{}$g_j(t - $\hat{{t}}$) E_j we arrive at

$${{\text{d}}u\over {\text{d}}t} {u\over \tau(t-\hat{t})} {1\over C} \hat{{t}} {1\over C}$$ (4.75)

which is a linear differential equation with spike-time dependent time constant; cf. Eq. (4.40). The effective time constant is shown in Fig. 4.21.

In step (ii) we integrate Eq. (4.75) with the initial condition

$$\hat{{t}} \Delta^{{\rm abs}}_{} \Delta^{{\rm abs}}_{} \tau \Delta^{{\rm abs}}_{}$$ (4.76)

and obtain

$$\eta \hat{{t}} \int_{0}^{{t-\hat{t}-\Delta^{\rm abs}}} \kappa \hat{{t}}$$ (4.77)

with

$$\kappa {1\over C} \left[\vphantom{-\int_{s-t}^{s} {{\text{d}}t' \over \tau(t')} }\right. \int_{{s-t}}^{{s}} {{\text{d}}t' \over \tau(t')} \left.\vphantom{-\int_{s-t}^{s} {{\text{d}}t' \over \tau(t')} }\right] \Theta \Delta^{{\rm abs}}_{} \Theta$$ (4.78)

$$\eta {1\over C} \int_{0}^{{s-\Delta^{\rm abs}}} \left[\vphantom{-\int_{s-t}^{s} {{\text{d}}t' \over \tau(t')} }\right. \int_{{s-t}}^{{s}} {{\text{d}}t' \over \tau(t')} \left.\vphantom{-\int_{s-t}^{s} {{\text{d}}t' \over \tau(t')} }\right] \Theta \Delta^{{\rm abs}}_{}$$ (4.79)

Figure 4.21: Variable time constant. The Spike Response Model with kernels (4.78) and (4.79) can be interpreted as an integrate-and-fire model with a time constant $\tau$ that depends on the time s since the last spike. Integration restarts after an absolute refractory period of 2ms with a time constant of 0.1ms. The time constant (solid line) relaxes first rapidly and then more slowly towards its equilibrium value of $\tau$ $\approx$ 4 ms (dashed line).

Finally, in step (iii) we introduce a dynamical threshold

$$\vartheta \left\{\vphantom{ \begin{array}{*{2}{c@{\qquad}}c} \vartheta^{\rm... ...rtheta} \right) \right] &{\rm for} &s \ge \Delta^{\rm abs} \end{array} }\right. \begin{array}{*{2}{c@{\qquad}}c} \vartheta^{\rm refr} &{\rm for} ... ... \tau_\vartheta} \right) \right] &{\rm for} &s \ge \Delta^{\rm abs} \end{array}$$ (4.80)

in order to fit the gain function for stationary input. During the absolute refractory period $\Delta^{{\rm abs}}_{}$ the threshold has been set to a value $\vartheta^{{\rm refr}}_{}$ = 100 mV that is sufficiently high to prevent the neuron from firing. After refractoriness, the threshold starts at zero and relaxes with a time constant of $\tau_{\vartheta}^{}$ = 6ms to an asymptotic value of $\vartheta_{0}^{}$ = - 50mV. With this set of parameters, we get a fair approximation of the gain function of the full cortical neuron model. The approximation for currents that are just super-threshold is bad, but for I^ext$\ge$5 $\mu$A/cm² the rates are not too far off, cf. Fig. 4.22A.

4.3.2.5 Example: Fluctuating input

We now test the Spike Response Model with the above set of parameters on a scenario with fluctuating input current. The mean firing rate of the full cortical neuron model and the Spike Response Model as a function of the fluctuation amplitude are similar; cf. Fig. 4.22B. Moreover, there is a high percentage of firing times of the Spike Response Model that coincide with those of the full model with a precision of $\Delta$ = ±2ms [coincidence rate $\Gamma$ = 0.75; cf. Eq. (4.67)]. A sample spike train is shown in Fig. 4.23A. Figure B exhibits a plot of the coincidence measure $\Gamma$ defined in equation (4.67) as a function of the fluctuation amplitude. In contrast to the nonlinear integrate-and-fire neuron, the coincidence rate is fairly constant over a broad range of stimulus amplitudes. At low rates, however, the coincidence rate drops off rapidly.

Figure 4.22: A. Stationary input. Gain function of the full model (solid line) and the Spike Response Model with constant (dotted line) and dynamic threshold (long-dashed line). B. Fluctuating input. The mean rate of the Spike Response Model (symbols) stimulated by random input is compared with that of the full model for the same input. The amplitude of the random input changes along the horizontal axis.

Figure 4.23: A. The spike train of the full model (solid line) is compared to that of the reduced model (dashed line). At about t = 1655 the reduced model misses a spike while it adds an extra spike about 10ms later. For this scenario about 80 percent of the spike times are correct within ±2 ms. B. Comparison of the coincidence rates $\Gamma$ for the multi-current integrate-and-fire model (dotted line), the nonlinear integrate-and-fire model (solid line), and the Spike Response Model (long-dashed line); cf. Fig. 4.20B.

4.3.3 Limitations

Not surprisingly, each approximation scheme is only valid in a limited regime. The natural question is thus whether this is the biologically relevant regime. Since a fluctuating input is probably the most realistic scenario, we have focused our discussion on this form of stimulation. We have seen that in case of a fluctuating input current, integrate-and-fire and Spike Response Model reproduce - to a certain extend - not only the mean firing rate, but also the firing times of the corresponding detailed neuron model. In this discipline, the multi-current integrate-and-fire model clearly yields the best performance. While it is easy to implement and rapid to simulate, it is difficult to analyze mathematically. Strictly speaking, it does not fall in the class of spiking neuron models reviewed in this chapter. It is interesting to see, however, that even the multi-current integrate-and-fire model which is based on a seemingly innocent approximation exhibits, for time-dependent input, a coincidence rate $\Gamma$ significantly below one. On the fluctuating-input task, we find that the single-variable (nonlinear) integrate-and-fire model exhibits a pronounced peak of $\Gamma$ at the optimal input, but does badly outside this regime. A Spike Response Model without adapting threshold yields coincidence rates that are not significantly different from the results for the nonlinear integrate-and-fire model. This indicates that the time-dependent threshold that has been included the definition of the Spike Response Model is an important component to achieve generalization over a broad range of different inputs. This also suggests that the nonlinear dependence of F(u) upon the membrane potential is not of eminent importance for the random-input task.

On the other hand, in the immediate neighborhood of the firing threshold, the nonlinear integrate-and-fire model performs better than the Spike Response Model. In fact, the Spike Response Model systematically fails to reproduce delayed action potentials triggered by input that is just slightly super-threshold. As we have seen, the nonlinear integrate-and-fire model is related to a canonical type I model and, therefore, exhibits the `correct' behavior in the neighborhood of the firing threshold.

In summary, it is always possible to design an input scenario where formal spiking neuron models fail. For example, none of the models discussed in this chapter is capable of reproducing effects of a slow adaptation to changes in the input.