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.
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 I0 plus a noise current Inoise. 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 fPSTH(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 - u0 between the mean membrane potential u0 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., fPSTH(t) (t); cf. Fig. 7.11B (top).
On the other hand, consider a neuron driven by a constant super-threshold current I0 without any noise. If an input spike arrives during the phase where the membrane potential u0(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., fPSTH(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 P0(s) A(t - s) in Eq. (7.3) vanishes and the dominant term is
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.
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) = I0 + I(t) that fluctuates around a mean value I0. 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
In this section, we want to relate the reverse correlation function Crev(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
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
For the sake of simplicity, we assume that the input consists of white noise7.1, i.e., the input has an autocorrelation
In order to relate the correlation function CAI to the reverse correlation Crev, we recall the definition of the population activity
|A(t) = (t - ti(f)) .||(7.71)|
We consider a SRM0 neuron u(t) = (t - ) + (s) I(t - s) ds 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) = I0 + I(t). The bias current I0 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 Crev(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
|0 = G(s) I(- s) ds + constP - I2(- s) ds||(7.78)|
|G(t) = 2 Iopt(- t)||(7.79)|
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