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 . If *u*_{i}(*t*) reaches
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
(*t* - *t*_{i}^{(f)}).

1.3.1 Definition of the Model SRM

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

where is the last firing time of neuron

The term in (1.3) describes the response of neuron

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
SRM_{0} 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 SRM_{0}
have a long tradition in theoretical neuroscience
(Hill, 1936; Stein, 1965; Weiss, 1966; Geisler and Goldberg, 1966).
Some important limitations of the model SRM_{0}
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.

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

with parameters ,,

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 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.

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*_{} 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

S_{i}(t) = (t - t_{i}^{(f)}) |
(1.6) |

where (

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 . Let us list the major limitations of this
approach.

**(i) Adaptation, Bursting, and Inhibitory Rebound**

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 *t*_{0} from a value *I*_{1} to a new value
*I*_{2}. Let us suppose that *I*_{1} = 0 so that the neuron is quiescent for *t* < *t*_{0}. If the current *I*_{2} is sufficiently large, it will evoke spikes for
*t* > *t*_{0}. 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 SRM_{0} 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 SRM_{0}
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 *I*_{1} < 0 and *I*_{2} = 0, i.e., an inhibitory input that is
switched off at time *t*_{0}; 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**

In the model SRM_{0} 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 *I*_{0}. We assume that *I*_{0} is too
weak to evoke firing so that, after some relaxation time, the membrane
potential settles at a constant value *u*_{0}. At *t* = *t*^{(f)} a presynaptic spike is
triggered. The spike generates a current pulse at the postsynaptic neuron
(postsynaptic current, PSC) with amplitude

PSC u_{0} - E_{syn} |
(1.7) |

where

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
*u*_{0} = *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
*u*_{0} *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.

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 and that a
presynaptic spike arrives at a time
*t*_{j}^{(f)} > . The form of the
postsynaptic potential depends now on the time
*t*_{j}^{(f)} - ;
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.

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 SRM_{0}. A purely linear dendrite, on the other hand, can be
incorporated in the model as we will see in Chapter 4.

Cambridge University Press, 2002

© Cambridge University Press

** This book is in copyright. No reproduction of any part
of it may take place without the written permission
of Cambridge University Press.**