moving stimuli is mediated by a correlation like operation which can be seen abstractly seen as a form of multiplication. Similar experimental observations exist for pigeons as well [28].
Finally Andersen [1] reviews some research papers, which indicate that “sensory signals from many modalities, converge in the posterior parietal cortex in order to code the spatial locations of goals for movement. These signals are combined using a specific gain mechanism. ”
3.4 Existing Models
In the literature there are some papers which propose models for multiplicative neural oper-ations. Most of these models are single cell’s specific biophysical mechanisms which could give rise to a multiplicative-like operation. However we should note that the research in this field is limited, despite the importance of understanding how multiplicative-like operations are neurally implemented. In our proposal we won’t concern with single cell models, but with small feed for-ward networks of Integrate-and-Fire neurons. On the following paragraphs we will briefly explain some of the multiplicative models found in literature.
3.4.1 Multiplication via Silent Inhibition
Silent inhibition in some special cases can give rise to a multiplicative behavior. We have seen in the previous chapter that synaptic current is given by:
Isyn(t) = gsyn(t) Vsynrev− Vm(t) . (3.3) If we suppose that the synaptic input changes slowly [18] then we can assume that the synaptic conductance gsyn(t) changes slowly with time. As a result there will be a stationary current and gsyn will be the constant synaptic input. If Rsyn the synaptic resistance then using Ohm’s law, V = R · I we take the following equation for the membrane voltage:
Vm = gsynRsynVsynrev
1 + RsynVsynrev. (3.4)
If the synaptic reversal is close to the resting potential of the cell (shunting inhibition) then the action of this synapse to Vm remains invisible.
From the previous equation we can take a multiplicative relation if we assume that the product of the synaptic resistance and synaptic conductance is small, gsynRsyn ≪ 1 :
Vm ≈ gsynRsynVsynrev. (3.5)
If we also have an excitatory synaptic input with an associated conductance change ge and a reversal potential Vexcrev then using Taylor expansion we take [18]:
Vm ≈ VexcrevRsyn ge− g2e− gegsynRsyn+ . . .
(3.6) which includes quadratic contributions from the excitatory synaptic terms and higher order terms from combinations of the excitatory and inhibitory inputs.
3.4.2 Spike Coincidence Detector
Srinivasan and Bernard [27] used an input spike coincidence detector in order to model mul-tiplication like responses. The main aim of the authors was not to model exact mulmul-tiplication
but to describe a scheme by which a neuron can produce a response which is proportional to the product of the input signals that it receives from two other neurons.
They investigated a neuronal model in which the neuron produces a spike only if it receives two spikes, from the two external neurons, that are coincident in time or nearly so. In Figure 3.4 we can see how such a neuron operates.
Figure 3.4 Neuron C receives input from two neurons A,B. Cell C fires a spike only if two input action potentials arrive within a ∆ ms. Only in this case the voltage membrane reaches the threshold. As a result the output firing rate of neuron C is proportional to the firing rates of A,B. (Figure taken from [27].)
In order to model coincidence detection, the proposed neuron spikes when its membrane voltage Vm is above a certain threshold Vthr. The presence of only one presynaptic spike cannot cause enough EPSP to discharge the cell, but if two spikes arrive within ∆ ms then the voltage threshold is reached and an output spike is generated. If Vmax the maximum membrane potential the neuron can reach from a single input spike, then there is an exponential decaying relation between membrane voltage and time,
V (t) = Vmaxe−τt (3.7)
where τ the neuron’s time constant. The authors make the assumption that Vmax< Vthr < 2Vmax
so a single spike cannot initiate a postsynaptic action potential. If we have an input spike at time t0 there should be another spike in an interval of ∆ ms before (or after) t0 ([t0 − ∆, t0 + ∆]) in order to have the postsynaptic action potential. Given two spikes at t and t + ∆ then the neuron will fire an action potential and as a result ∆ can easily be determined by the equation:
Vmaxe−∆τ + Vmax = Vthr. (3.8)
The authors assumed statistical independence of the two input firing rates (a natural assump-tion in most cases, for example when the stimuli causing activaassump-tion of the two presynaptic cells
Section 3.4 - Existing Models 19 are independent) and showed that the output firing rate is proportional to the product of the two input firing frequencies [27]:
fout = 2∆fAfB (3.9)
Chapter 4
Multiplication with Networks of I&F Neurons
4.1 Introduction
In the previous chapter we presented evidence of multiplicative behavior in neural cells. We also argued for the importance of this simple nonlinear operation. Despite its simplicity it is unclear how biological neural networks implement multiplication. Also the research done in this field is limited and the models found in bibliography (we presented some of them in the previous chapter), are complex single cell biophysical mechanisms.
We try to approach multiplication using very simple networks of Integrate-and-Fire neurons and a combination of excitatory and inhibitory synapses. In this chapter we are going to present the underlying theory and the proposed models. We also analyze in depth the main idea behind this dissertation which is the usage of the minimum function for implementing a neural multiplicative operator.