Subsections

• 1.3.1 Definition of the Model SRM₀
  • 1.3.1.1 Example: Formal pulses
  • 1.3.1.2 Example: Formal spike trains
  
  
• 1.3.2 Limitations of the Model
  • 1.3.2.1 Example: Shunting Inhibition and Reversal Potential
  • 1.3.2.2 Example: Conductance Changes after a Spike
  • 1.3.2.3 Example: Spatial Structure

1.3 A Phenomenological Neuron Model

In order to build a phenomenological model of neuronal dynamics, we describe the critical voltage for spike initiation by a formal threshold $\vartheta$. If u_i(t) reaches $\vartheta$ from below we say that neuron i fires a spike. The moment of threshold crossing defines the firing time t_i^(f). The model makes use of the fact that action potentials always have roughly the same form. The trajectory of the membrane potential during a spike can hence be described by a certain standard time course denoted by $\eta$(t - t_i^(f)).

1.3.1 Definition of the Model SRM₀

Putting all elements together we have the following description of neuronal dynamics. The variable u_i describes the momentary value of the membrane potential of neuron i. It is given by

$$\eta \hat{{t}}_{i}^{} \sum_{j}^{} \sum_{{f}}^{} \epsilon_{{ij}}^{}$$ (1.3)

where $\hat{{t}}_{i}^{}$ is the last firing time of neuron i, i.e., $\hat{{t}}_{i}^{}$ = max{t_i^(f) | t_i^(f) < t}. Firing occurs whenever u_i reaches the threshold $\vartheta$ from below,

$$\vartheta {{\rm d}\over {\rm d}t} \Longrightarrow$$ (1.4)

The term $\epsilon_{{ij}}^{}$ in (1.3) describes the response of neuron i to spikes of a presynaptic neuron j. The term $\eta$ in (1.3) describes the form of the spike and the spike-afterpotential.

Note that we are only interested in the potential difference, viz., the distance from the resting potential. By an appropriate shift of the voltage scale, we can always set u_rest = 0. The value of u(t) is then directly the distance from the resting potential. This is implicitly assumed in most neuron models discussed in this book.

The model defined in equations (1.3) and (1.4) is called SRM₀ where SRM is short for Spike Response Model (Gerstner, 1995). The subscript zero is intended to remind the reader that it is a particularly simple `zero order' version of the full model that will be introduced in Chapter 4. Phenomenological models of spiking neurons similar to the models SRM₀ have a long tradition in theoretical neuroscience (Hill, 1936; Stein, 1965; Weiss, 1966; Geisler and Goldberg, 1966). Some important limitations of the model SRM₀ are discussed below in Section 1.3.2. Despite the limitations, we hope to be able to show in the course of this book that spiking neuron models such as the Spike Response Model are a useful conceptual framework for the analysis of neuronal dynamics and neuronal coding.

1.3.1.1 Example: Formal pulses

In a simple model, we may replace the exact form of the trajectory $\eta$ during an action potential by, e.g., a square pulse, followed by a negative spike-afterpotential,

$$\eta \left\{\vphantom{ \begin{array}{*{6}{c@{\,}}c} {1/ \Delta t} \qqu... ...ver \tau}\right) \quad&{\rm for}& \Delta t&<& t-t_i^{(f)}& \end{array} }\right. \begin{array}{*{6}{c@{\,}}c} {1/ \Delta t} \qquad &{\rm for}& 0&<... ...i^{(f)}\over \tau}\right) \quad&{\rm for}& \Delta t&<& t-t_i^{(f)}& \end{array}$$ (1.5)

with parameters $\eta_{0}^{}$,$\tau$,$\Delta$t > 0. In the limit of $\Delta$t$\to$ 0 the square pulse approaches a Dirac $\delta$ function; see Fig. 1.4.

The positive pulse marks the moment of spike firing. For the purpose of the model, it has no real significance, since the spikes are recorded explicitly in the set of firing times t_i⁽¹⁾, t_i⁽²⁾,.... The negative spike-afterpotential, however, has an important implication. It leads after the pulse to a `reset' of the membrane potential to a value below threshold. The idea of a simple reset of the variable u_i after each spike is one of the essential components of the integrate-and-fire model that will be discussed in detail in Chapter 4.

If $\eta_{0}^{}$ $\gg$ $\vartheta$ then the membrane potential after the pulse is significantly lower than the resting potential. The emission of a second pulse immediately after the first one is therefore more difficult, since many input spikes are needed to reach the threshold. The negative spike-after potential in Eq. (1.5) is thus a simple model of neuronal refractoriness.

Figure 1.4: In formal models of spiking neurons the shape of an action potential (dashed line) is usually replaced by a $\delta$ pulse (vertical line). The negative overshoot (spike-afterpotential) after the pulse is included in the kernel $\eta$(t - t_i⁽¹⁾) (thick line) which takes care of `reset' and `refractoriness'. The pulse is triggered by the threshold crossing at t_i⁽¹⁾. Note that we have set u_rest = 0.

1.3.1.2 Example: Formal spike trains

Throughout this book, we will refer to the moment when a given neuron emits an action potential as the firing time of that neuron. In models, the firing time is usually defined as the moment of threshold crossing. Similarly, in experiments firing times are recorded when the membrane potential reaches some threshold value u_$\vartheta$ from below. We denote firing times of neuron i by t_i^(f) where f = 1, 2,... is the label of the spike. Formally, we may denote the spike train of a neuron i as the sequence of firing times

$$\sum_{f}^{} \delta$$ (1.6)

where $\delta$(x) us the Dirac $\delta$ function with $\delta$(x) = 0 for x$\ne$ 0 and $\int_{{-\infty}}^{\infty}$$\delta$(x)dx = 1. Spikes are thus reduced to points in time.

1.3.2 Limitations of the Model

The model presented in Section 1.3.1 is highly simplified and neglects many aspects of neuronal dynamics. In particular, all postsynaptic potentials are assumed to have the same shape, independently of the state of the neuron. Furthermore, the dynamics of neuron i depends only on its most recent firing time $\hat{{t}}_{i}^{}$. Let us list the major limitations of this approach.

(i) Adaptation, Bursting, and Inhibitory Rebound

Figure 1.5: Response to a current step. In A - C, the current is switched on at t = t₀ to a value I₂ > 0. Regular-spiking neurons (A) exhibit adaptation of the interspike intervals whereas fast-spiking neurons (B) show no adaptation. An example of a bursting neuron is shown in C. Many neurons emit an inhibitory rebound spike (D) after an inhibitory current I₁ < 0 is switched off. Schematic figure.

To study neuronal dynamics experimentally, neurons can be isolated and stimulated by current injection through an intracellular electrode. In a standard experimental protocol we could, for example, impose a stimulating current that is switched at time t₀ from a value I₁ to a new value I₂. Let us suppose that I₁ = 0 so that the neuron is quiescent for t < t₀. If the current I₂ is sufficiently large, it will evoke spikes for t > t₀. Most neurons will respond to the current step with a spike train where intervals between spikes increase successively until a steady state of periodic firing is reached; cf. Fig. 1.5A. Neurons that show this type of adaptation are called regularly-firing neurons (Connors and Gutnick, 1990). Adaptation is a slow process that builds up over several spikes. Since the model SRM₀ takes only the most recent spike into account, it cannot capture adaptation. Detailed neuron models which will be discussed in Chapter 2 describe the slow processes that lead to adaptation explicitly. To mimic adaptation with formal spiking neuron models we would have to add up the contributions to refractoriness of several spikes back in the past; cf. Chapter 4.

A second class of neurons are fast-spiking neurons. These neurons show now adaptation and can therefore be well approximated by the model SRM₀ introduced in Section 1.3.1. Many inhibitory neurons are fast-spiking neurons. Apart from regular-spiking and fast-spiking neurons, there are also bursting neurons which form a separate group (Connors and Gutnick, 1990). These neurons respond to constant stimulation by a sequence of spikes that is periodically interrupted by rather long intervals; cf. Fig. 1.5C. Again, a neuron model that takes only the most recent spike into account cannot describe bursting. For a review of bursting neuron models, the reader is referred to (Izhikevich, 2000).

Another frequently observed behavior is post-inhibitory rebound. Consider a step current with I₁ < 0 and I₂ = 0, i.e., an inhibitory input that is switched off at time t₀; cf. Fig. 1.5D. Many neurons respond to such a change with one or more `rebound spikes': Even the release of inhibition can trigger action potentials. We will return to inhibitory rebound in Chapter 2.

(ii) Saturating excitation and shunting inhibition

Figure 1.6: The shape of postsynaptic potentials depends on the momentary level of depolarization. A. A presynaptic spike that arrives at time t^(f) at an inhibitory synapse has hardly any effect on the membrane potential when the neuron is at rest, but a large effect if the membrane potential u is above the resting potential. If the membrane is hyperpolarized below the reversal potential of the inhibitory synapse, the response to the presynaptic input changes sign. B. A spike at an excitatory synapse evokes a postsynaptic potential with an amplitude that depends only slightly on the momentary voltage u. For large depolarizations the amplitude becomes smaller (saturation). Schematic figure.

In the model SRM₀ introduced in Section 1.3.1, the form of a postsynaptic potential generated by a presynaptic spike at time t_j^(f) does not depend on the state of the postsynaptic neuron i. This is of course a simplification and reality is somewhat more complicated. In Chapter 2 we will discuss detailed neuron models that describe synaptic input as a change of the membrane conductance. Here we simply summarize the major phenomena.

In Fig. 1.6 we have sketched schematically an experiment where the neuron is driven by a constant current I₀. We assume that I₀ is too weak to evoke firing so that, after some relaxation time, the membrane potential settles at a constant value u₀. At t = t^(f) a presynaptic spike is triggered. The spike generates a current pulse at the postsynaptic neuron (postsynaptic current, PSC) with amplitude

$$\propto$$ (1.7)

where u₀ is the membrane potential and E_syn is the `reversal potential' of the synapse. Since the amplitude of the current input depends on u₀, the response of the postsynaptic potential does so as well. Reversal potentials are systematically introduced in Chapter 2.2; models of synaptic input are discussed in Chapter 2.4.

1.3.2.1 Example: Shunting Inhibition and Reversal Potential

The dependence of the postsynaptic response upon the momentary state of the neuron is most pronounced for inhibitory synapses. The reversal potential of inhibitory synapses E_syn is below, but usually close to the resting potential. Input spikes thus have hardly any effect on the membrane potential if the neuron is at rest; cf. 1.6a. However, if the membrane is depolarized, the very same input spikes evoke a nice inhibitory postsynaptic potentials. If the membrane is already hyperpolarized, the input spike can even produce a depolarizing effect. There is a intermediate value u₀ = E_syn - the reversal potential - where the response to inhibitory input `reverses' from hyperpolarizing to depolarizing.

Though inhibitory input usually has only a small impact on the membrane potential, the local conductivity of the cell membrane can be significantly increased. Inhibitory synapses are often located on the soma or on the shaft of the dendritic tree. Due to their strategic position a few inhibitory input spikes can `shunt' the whole input that is gathered by the dendritic tree from hundreds of excitatory synapses. This phenomenon is called `shunting inhibition'.

The reversal potential for excitatory synapses is usually significantly above the resting potential. If the membrane is depolarized u₀ $\gg$ u_rest the amplitude of an excitatory postsynaptic potential is reduced, but the effect is not as pronounced as for inhibition. For very high levels of depolarization a saturation of the EPSPs can be observed; cf. 1.6b.

1.3.2.2 Example: Conductance Changes after a Spike

Figure 1.7: The shape of postsynaptic potentials (dashed lines) depends on the time t - $\hat{{t}}_{i}^{}$ that has passed since the last output spike current if neuron i. The postsynaptic spike has been triggered at time $\hat{{t}}_{i}^{}$. A presynaptic spike that arrives at time t_j^(f) shortly after the spike of the postsynaptic neuron has a smaller effect than a spike that arrives much later. The spike arrival time is indicated by an arrow. Schematic figure.

The shape of the postsynaptic potentials does not only depend on the level of depolarization but, more generally, on the internal state of the neuron, e.g., on the timing relative to previous action potentials.

Suppose that an action potential has occured at time $\hat{{t}}_{i}^{}$ and that a presynaptic spike arrives at a time t_j^(f) > $\hat{{t}}_{i}^{}$. The form of the postsynaptic potential depends now on the time t_j^(f) - $\hat{{t}}_{i}^{}$; cf. Fig. 1.7. If the presynaptic spike arrives during or shortly after a postsynaptic action potential it has little effect because some of the ion channels that were involved in firing the action potential are still open. If the input spike arrives much later it generates a postsynaptic potential of the usual size. We will return to this effect in Chapter 2.2.

1.3.2.3 Example: Spatial Structure

The form of postsynaptic potentials also depends on the location of the synapse on the dendritic tree. Synapses that are located at the distal end of the dendrite are expected to evoke a smaller postsynaptic response at the soma than a synapse that is located directly on the soma; cf. Chapter 2. If several inputs occur on the same dendritic branch within a few milliseconds, the first input will cause local changes of the membrane potential that influence the amplitude of the response to the input spikes that arrive slightly later. This may lead to saturation or, in the case of so-called `active' currents, to an enhancement of the response. Such nonlinear interactions between different presynaptic spikes are neglected in the model SRM₀. A purely linear dendrite, on the other hand, can be incorporated in the model as we will see in Chapter 4.