- 7.4.1 The Effect of an Input Spike
- 7.4.1.1 Example: The input-output crosscorrelation of integrate-and-fire neurons
- 7.4.1.2 Example: Input-output measurements in motoneurons

- 7.4.2 Reverse Correlation - the Significance of an Output Spike

7.4 The Significance of a Single Spike

The above results derived for a *population* of spiking neurons have an
intimate relation to experimental measurements of the input-output transforms
of a *single* neuron as typically measured by a peri-stimulus time
histogram (PSTH) or by reverse correlations. This relation allows to give an
interpretation of population results in the language of neural coding; see
Chapter 1.4. In particular, we would like to understand the
`meaning' of a spike. In Section 7.4.1 we focus on the
typical effect of a single *presynaptic* spike on the firing probability
of a postsynaptic neuron. In Section 7.4.2 we study how
much we can learn from a single *postsynaptic* spike about the
presynaptic input.

7.4.1 The Effect of an Input Spike

What is the typical response of a neuron to a single presynaptic spike? An
experimental approach to answer this question is to study the temporal
response of a single neuron to current pulses (Fetz and Gustafsson, 1983; Poliakov et al., 1997). More
precisely a neuron is driven by a constant background current *I*_{0} plus a
noise current
*I*_{noise}. At time *t* = 0 an additional short current
pulse is injected into the neuron that mimics the time course of an excitatory
or inhibitory postsynaptic current. In order to test whether this extra input
pulse can cause a postsynaptic action potential the experiment is repeated
several times and a peri-stimulus time histogram (PSTH) is compiled. The PSTH
can be interpreted as the probability density of firing as a function of time
*t* since the stimulus, here denoted
*f*_{PSTH}(*t*). Experiments show
that the shape of the PSTH response to an input pulse is determined by the
amount of synaptic noise and the time course of the postsynaptic potential
(PSP) caused by the current pulse
(Kirkwood and Sears, 1978; Moore et al., 1970; Knox, 1974; Fetz and Gustafsson, 1983; Poliakov et al., 1997).

How can we understand the relation between
postsynaptic potential and PSTH?
There are two different intuitive pictures; cf.
Fig. 7.11.
First, consider a neuron driven by stochastic background input.
If the input is not too strong, its membrane potential *u*
hovers somewhere below threshold. The shorter the distance
- *u*_{0}
between the mean membrane potential *u*_{0} and the
threshold
the higher the probability
that the fluctuations drive the neuron to firing.
Let us suppose that at
*t* = 0 an additional excitatory input spike arrives. It causes
an excitatory postsynaptic potential with time course
(*t*) which drives the
mean potential closer to threshold. We
therefore expect (Moore et al., 1970)
that the probability density
of firing (and hence the PSTH) shows a time course
similar to
the time course of the postsynaptic potential,
i.e.,
*f*_{PSTH}(*t*) (*t*);
cf. Fig. 7.11B (top).

On the other hand, consider a neuron driven
by a constant super-threshold current *I*_{0}
without any noise.
If an input spike arrives during the phase where the membrane potential
*u*_{0}(*t*) is just below threshold,
it may trigger a spike. Since the threshold crossing
can only occur during the rising phase
of the postsynaptic potential,
we may expect (Kirkwood and Sears, 1978) that the PSTH is proportional
to the *derivative* of the postsynaptic potential, i.e.,
*f*_{PSTH}(*t*) (*t*);
cf. Fig. 7.11B (bottom).

Both regimes can be observed in simulations of integrate-and-fire neurons; cf.
Fig. 7.12. An input pulse at *t* = 0 causes a PSTH. The shape
of the PSTH depends on the noise level and is either similar to the
postsynaptic potential or to its derivative. Closely related effects have
been reported in the experimental literature cited above. In this section we
show that the theory of signal transmission by a population of spiking neurons
allows us to analyze these results from a systematic point of view.

In order to understand how the theory of population activity
can be applied to single-neuron PSTHs,
let us consider a homogeneous population of *N* unconnected,
noisy neurons initialized with
random initial conditions, all receiving the same input.
Since the neurons are independent,
the activity of the population as a whole in response to a given stimulus
is equivalent to the PSTH compiled from the response of a single
noisy neuron to *N* repeated presentations of the same stimulus.
Hence, we can apply theoretical results for the activity of
homogeneous populations to the PSTH of an individual neuron.

Since a presynaptic spike causes typically an input pulse of small amplitude,
we may calculate the PSTH from the linearized population activity equation;
cf. Eq. (7.3). During the *initial* phase of the response,
the integral over
*P*_{0}(*s*) *A*(*t* - *s*) in Eq. (7.3) vanishes
and the dominant term is

where (

7.4.1.1 Example: The input-output crosscorrelation of integrate-and-fire neurons

In this example we study integrate-and-fire neurons with escape noise. A bias
current is applied so that we have a constant baseline firing rate of about
30Hz. At *t* = 0 an excitatory (or inhibitory) current pulse is applied which
increases (or decreases) the firing density as measured with the PSTH;
cf. Fig. 7.13.
At low
noise the initial response is followed by a decaying oscillation with a period
equal to the single-neuron firing rate. At high noise the response is
proportional to the excitatory (or inhibitory) postsynaptic potential. Note
the asymmetry
between excitation and inhibition,
i.e., an the response to an inhibitory current
pulse is smaller than that to an excitatory one.
The linear theory
can not reproduce this asymmetry.
It is, however, possible to integrate the full nonlinear
population equation (6.75) using the methods discussed in
Chapter 6. The numerical integration reproduces nicely
the non-linearities found in the simulated PSTH;
cf. Fig. 7.13A.

In this example we compare theoretical results with experimental input-output
measurements in motoneurons (Fetz and Gustafsson, 1983; Poliakov et al., 1996,1997). In the study
of Poliakov et al. (1997), PSTH responses to Poisson-distributed trains of
current pulses were recorded. The pulses were injected into the soma of rat
hypoglossal motoneurons during repetitive discharge. The time course of the
pulses was chosen to mimic postsynaptic currents generated by presynaptic
spike arrival. PSTHs of motoneuron discharge occurrences were compiled when
the pulse trains were delivered either with or without additional current
noise which simulated noisy background input. Fig. 7.14
shows examples of responses from a rat motoneuron taken from the work of
Poliakov which is a continuation of earlier work
(Moore et al., 1970; Kirkwood and Sears, 1978; Knox, 1974; Fetz and Gustafsson, 1983). The effect of adding noise can be
seen clearly: the low-noise peak is followed by a marked trough, whereas the
high-noise PSTH has a reduced amplitude and a much smaller trough. Thus, in
the low-noise regime (where the type of noise model is irrelevant) the
response to a synaptic input current pulse is similar to the * derivative* of the postsynaptic potential (Fetz and Gustafsson, 1983), as predicted by
earlier theories (Knox, 1974), while for high noise it is similar to the
postsynaptic potential itself.

Fig. 7.14C and D shows PSTHs produced by a Spike Response Model of a motoneuron; cf. Chapter 4.2. The model neuron is stimulated by exactly the same type of stimulus that was used in the above experiments on motoneurons. The simulations of the motoneuron model are compared with the PSTH response predicted from the theory. The linear response reproduces the general characteristics that we see in the simulations. The full nonlinear theory derived from the numerical solution of the population equation fits nicely with the simulation. The results are also in qualitative agreement with the experimental data.

7.4.2 Reverse Correlation - the Significance of an Output Spike

In a standard experimental protocol to characterize the coding
properties of a single neuron, the neuron
is driven by a time-dependent stimulus
*I*(*t*) = *I*_{0} + *I*(*t*)
that fluctuates around
a mean value *I*_{0}. Each time the neuron emits a spike, the time-course of the input just before
the spike is recorded. Averaging over many spikes yields the typical input
that drives the neuron towards firing. This spike-triggered average is called
the `reverse correlation' function; cf. Chapter 1.4. Formally,
if neuronal firing times are denoted by *t*^{(f)} and the stimulus before the
spike by
*I*(*t*^{(f)} - *s*), we define the reverse correlation function as

where the average is to be taken over all firing times

In this section, we want to relate the reverse correlation
function
*C*^{rev}(*s*)
to the signal transfer properties of a single neuron
(Bryant and Segundo, 1976).
In Section 7.3,
we have seen that, in the linear regime,
signal transmission properties of a population of neuron
are described by

with a frequency-dependent gain (); see Eq. (7.57). We will use that, for independent neurons, the transfer characteristics of a population are identical to that of a single neuron. We therefore interpret () as the single-neuron transfer function. Inverse Fourier transform of Eq. (7.69) yields

with a kernel defined in Eq. (7.59).

Eq. (7.70)
describes the relation between a known (deterministic)
input
*I*(*t*)
and the population activity.
We now adopt a statistical point of view
and assume that the input
*I*(*t*) is drawn from a statistical
ensemble of stimuli with mean
*I*(*t*) = 0.
Angular brackets denote averaging over the input
ensemble or, equivalently, over an infinite input sequence.
We are interested in the correlation

between input

where we have used

For the sake of simplicity, we assume that the input
consists of white noise^{7.1}, i.e., the input has an autocorrelation

In this case Eq. (7.72) reduces to

Thus the correlation function

In order to relate the correlation function *C*_{AI}
to the reverse correlation
*C*^{rev},
we recall the definition of the population activity

A(t) = (t - t_{i}^{(f)}) . |
(7.71) |

The correlation function (7.71) is therefore

Thus the value of the correlation function

Since we have focused on a population of independent neurons, the reverse correlation of the population is identical to that of a

This is an important result. For spiking neurons the transfer function

We consider a SRM_{0} neuron
*u*(*t*) = (*t* - ) + (*s*) *I*(*t* - *s*) d*s*
with piecewise linear
escape noise. The response kernels are exponential
with a time constant of = 4ms for the
kernel and
= 20 ms
for the refractory kernel .
The neuron is driven by a current
*I*(*t*) = *I*_{0} + *I*(*t*).
The bias current *I*_{0} was adjusted so that
the neuron fires at a mean rate of 50Hz.
The noise current was generated by the following procedure.
Every time step of 0.1ms we apply with a probability
of 0.5 an input pulse. The amplitude of the pulse
is ±1 with equal probability.
To estimate the reverse correlation function,
we build up a histogram of the average
input
*I*(*t* - *t*^{(f)}) preceding
a spike *t*^{(f)}.
We see from Fig. 7.15A
that
the main characteristics
of the reverse correlation function are already visible
after 1000 spikes.
After an average over 25000 spikes, the time course
is much cleaner and reproduces to a high degree
of accuracy the
time course of the time-reversed
impulse response *G*(- *s*) predicted by the theory;
cf. Fig. 7.15B.
The oscillation with a period of about 20ms
reflects the intrinsic firing period of the neuron.

In this example we want to show that the reverse correlation function
*C*^{rev}(*s*)
can be interpreted as the optimal stimulus to trigger a spike.
To do so, we assume that the amplitude of the stimulus
is small and use the linearized population equation

Suppose that we want to have a large response

which one will give the maximal response

To prove the assertion, we need to maximize

0 = G(s) I(- s) ds + const_{P} - I^{2}(- s) ds |
(7.78) |

which must hold at any arbitrary time

G(t) = 2 I_{opt}(- t) |
(7.79) |

which proves the assertion (7.81). The exact value of could be determined from Eq. (7.80) but is not important for our arguments. Finally, from Eq. (7.78) we have

Cambridge University Press, 2002

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