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12.4 Subtraction of Expectations

In this and the following section we discuss temporal coding in specialized neuronal systems. We start in this section with the problem of electro-sensory signal processing in Mormoryd electric fish.

12.4.1 Electro-Sensory System of Mormoryd Electric Fish

Mormoryd electric fish probe their environment with electric pulses. The electric organ of the fish emits a short electric discharge. The spatio-temporal electric field that is generated around the fish by the discharge depends on the location of objects in the surroundings. In order to reliably detect the location, size, or movement of objects the electro-sensory system must compare the momentary spatio-temporal electric field with the one that would occur in the absence of external objects. In other words, it must subtract the expected spatio-temporal image from the actual sensory input.

Experiments have shown that so-called medium ganglion cells in the electro-sensory lateral lobe (ELL) of electric fish can solve the task of subtracting expectations (Bell et al., 1997a). These cells receive two sets of input; cf. Fig. 12.10A. Information on the timing of a discharge pulse emitted by the electric organ is conveyed via a set of delay lines to the ganglion cells. The signal transmission delay $ \Delta_{j}^{}$ between the electric organ discharge and spike arrival at the ganglion cell changes from one connection to the next and varies between zero and 100milliseconds. A second set of input conveys the characteristics of the spatio-temporal electric field sensed by the fish's electro-receptors.

Figure 12.10: A. Schematic view of ganglion cell input. Information about the timing of electric organ discharge is transmitted by a set of delayed spikes (top) while the sensory image is encoded in a second set of spikes (bottom). B. Synaptic plasticity as a function of the time difference s = tj(f) - ti(f) between postsynaptic broad spikes and presynaptic spikes. The synapse is depressed, if the presynaptic spike arrives slightly before the back-propagating action potential (-100ms< s < 0) while it is strongly increased if the presynaptic spike arrives slightly after the postsynaptic one. This dependence is modeled by a spike-time dependent learning window W(s). The presynaptic term a1pre (horizontal dashed line) takes care of the fact that for very large time differences there is always a positive synaptic change. Experimental data points redrawn after Bell et al. (1997b).
\hbox{{\bf A} \hspace{53mm} {\bf B}}
\includegraphics[height=40mm]{Figs-ch-hebbcode/CCBell.eps} }

In experiments, electric organ discharges are triggered repetitively at intervals of T = 150 ms. If the sensory input has, after each discharge, the same spatio-temporal characteristics, the ganglion cell responds with stochastic activity at a constant rate; cf. Fig. 12.11A. If the sensory input suddenly changes, the ganglion cell reacts strongly. Thus the ganglion cell can be seen as a novelty detector. The predictable contribution of the sensory image is subtracted, and only unpredictable aspects of a sensory image evoke a response. In the following paragraph we will show that a spike-time dependent learning rule with anti-Hebbian characteristics can solve the task of subtracting expectations (Roberts and Bell, 2000; Bell et al., 1997b).

12.4.2 Sensory Image Cancellation

In this section we review the model of Roberts and Bell (2000). We start with the model of the ganglion cell, turn then to the model of synaptic plasticity, and compare finally the model results with experimental results of ganglion cell activity. Neuron model

We consider a single ganglion cell that receives two sets of inputs as indicated schematically in Fig. 12.10A. After each electric organ discharge, a volley of 150 input spikes arrives at different delays $ \Delta_{1}^{}$, $ \Delta_{2}^{}$...$ \Delta_{{150}}^{}$. Each spike evokes upon arrival an excitatory postsynaptic potential with time course $ \epsilon$(s). A second set of input carries the sensory stimulus. Instead of modeling the sequence of spike arrival times, the time course of the stimulus is summarized by a function hstim(s) where s = 0 is the moment of the electric organ discharge. The total membrane potential of the ganglion cell i is

ui(t) = $\displaystyle \sum_{{n=1}}^{{n_{\rm max}}}$$\displaystyle \left[\vphantom{ \sum_j w_{ij} \, \epsilon(t-n\,T-\Delta_j) + h^{\rm stim}(t-n\,T) }\right.$$\displaystyle \sum_{j}^{}$wij $\displaystyle \epsilon$(t - n T - $\displaystyle \Delta_{j}^{}$) + hstim(t - n T)$\displaystyle \left.\vphantom{ \sum_j w_{ij} \, \epsilon(t-n\,T-\Delta_j) + h^{\rm stim}(t-n\,T) }\right]$ (12.8)

where wij is the synaptic efficacy, T = 150 ms the interval between electric organ discharges, and nmax the total number of repetitions of the stimulus hstim up to time t.

A ganglion cell is described as a (semi-)linear Poisson model that emits spikes at a rate

$\displaystyle \nu_{i}^{}$(t) = [ui(t) - $\displaystyle \vartheta$]+ . (12.9)

There are two types of action potential, i.e., a narrow spike that travels along the axon and transmits information to other neurons; and a broad spike that back-propagates into the apical dendrite. It is therefore the broad spike that conveys information on postsynaptic spike firing to the site of the synapse. Both types of spike are generated by an inhomogeneous Poisson process with rate (12.11) but the threshold $ \vartheta$ of the broad spike is higher than that of the narrow spike.

Figure 12.11: Sensory image cancellation and negative after-image in experiments (A) and simulation (B). Each trial corresponds to one horizontal line. The electric organ discharge is elicited at time t = 0; Spikes are denoted by black dots. For the first 50 trials no sensory input is given. After trial 50 a time-dependent stimulus is applied which is repeated in the following trials. The neuron adapts. After the stimulus is removed (trial 3500), the time course of the neuronal activity exhibits a negative after-image of the stimulus. The stimulus onset is denoted by a vertical bar; taken with slight adaptations from Roberts and Bell (2000), experiments of Bell et al. (1997a).
\hspace{30mm}{\bf A} \hspace{35mm} {\bf B}
\par\vspace{-80mm} Synaptic plasticity

The synaptic efficacy wij changes according to a spike-time dependent plasticity rule

$\displaystyle {\frac{{{\text{d}}}}{{{\text{d}}t}}}$wij(t) = Sj(t$\displaystyle \left[\vphantom{ a_1^{\text{pre}} + \int_0^\infty W(s) \, S_i(t-s) \; {\text{d}}s }\right.$a1pre + $\displaystyle \int_{0}^{\infty}$W(sSi(t - s)  ds$\displaystyle \left.\vphantom{ a_1^{\text{pre}} + \int_0^\infty W(s) \, S_i(t-s) \; {\text{d}}s }\right]$    
  + Si(t$\displaystyle \left[\vphantom{ \int_0^\infty W(-s) \, S_j(t-s) \; {\text{d}}s }\right.$$\displaystyle \int_{0}^{\infty}$W(- sSj(t - s)  ds$\displaystyle \left.\vphantom{ \int_0^\infty W(-s) \, S_j(t-s) \; {\text{d}}s }\right]$ ; (12.10)

cf. Eq. (10.14). Here a1pre > 0 is a non-Hebbian presynaptic term, W(s) with $ \bar{W}$ = $ \int_{{-\infty}}^{\infty}$W(s) ds < 0 is the learning window, Sj(t) = $ \sum_{f}^{}$$ \delta$(t - tj(f)) is the train of presynaptic spike arrivals, and Si(t) = $ \sum_{f}^{}$$ \delta$(t - ti(f)) is the train of postsynaptic broad spikes. The learning window W(ti(f) - tj(f)) as a function of the time difference between a back-propagating broad spike and a presynaptic spike has been measured experimentally (Bell et al., 1997b). It has two phases, one for LTP and one for LTD, but the timing is reversed as compared to other spike-time dependent plasticity rules (Zhang et al., 1998; Markram et al., 1997; Bi and Poo, 1998; Debanne et al., 1998). In particular, a presynaptic spike that arrives slightly before a postsynaptic broad spike leads to depression of the synapse; cf. Fig. 12.10B. It can thus be called anti-Hebbian. While Hebbian learning windows are apt to detect and reinforce temporal structure in the input, anti-Hebbian rules suppress any temporal structure, as we will see in the following paragraph. Results

A simulation of the model introduced above is shown in Fig. 12.11B. During the first 50 trials, no stimulus was applied ( hstim $ \equiv$ 0). In all subsequent trials up to trial 3500, an inhibitory stimulus hstim(s) with triangular time course has been applied. While the activity is clearly suppressed in trial 51, it recovers after several hundred repetitions of the experiment. If the stimulus is removed thereafter, the neuronal activity exhibits a negative after-image of the stimulus, just as in the experiments shown in Fig. 12.11A.

Using the methods developed in Chapter 11, it is possible to show that, for a1pre > 0 and $ \bar{W}$ < 0, the learning rule stabilizes the mean output rate (of broad spikes) at a fixed point $ \nu_{{\text{FP}}}^{}$ = - a1pre/$ \bar{W}$. Moreover, weights wij are adjusted so that the membrane potential has minimal fluctuations about uFP = $ \nu_{{\text{FP}}}^{}$ + $ \vartheta$. To achieve this, the weights must be tuned so that the term $ \sum_{j}^{}$wij $ \epsilon$(t - $ \Delta_{j}^{}$) cancels the sensory input hstim(t) - which is the essence of sensory image cancellation (Roberts and Bell, 2000).

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Next: 12.5 Transmission of Temporal Up: 12. Plasticity and Coding Previous: 12.3 Sequence Learning
Gerstner and Kistler
Spiking Neuron Models. Single Neurons, Populations, Plasticity
Cambridge University Press, 2002

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