## 2 Background
### Biologically Inspired Neuron Models  
The model presented below was originally introduced by [PERSON] and [PERSON] to investigate the behavior of a giant squid axon [11]. It has been adopted by many researchers as the benchmark model for neural activity. Figure 1 shows the equivalent electrical circuits for the Hodgkin-Huxley (HH) model and the Leaky Integrate-and-Fire (LIF) neuron model. One can see the HH model can be reduced to the LIF model by removing the two branches of circuit representing the conductance of Na and K ion channels. In this circuit, \(C\) is the capacitance per unit area of the lipid bi-layer (membrane) separating the ions on the inside and the outside of the cell. The fixed resistance symbol indicates the current for a non-specific leak. Variable resistances indicate the voltage dependence of the conductance on the Na\({}^{+}\) and K\({}^{+}\) ions. The _emf_ indicates the driving force for the ions, which is given in the model by the difference between membrane potential, \(V=V_{in}-V_{out}\) and reversal potential. \(I\) is the total membrane current per unit area, \(G_{K}\) and G\({}_{Na}\) are the potassium conductance per unit area and sodium conductance per unit area, respectively. \(V_{K}\) and \(V_{Na}\) are the potassium and sodium reverse potentials, respectively. \(G_{L}\) and \(V_{L}\) are the leak conductance per unit area, and leak reverse voltage, respectively. The action potential of the axon follows the equations:  
\[C\frac{dV}{dt}=I-G_{K}n^{4}(V-V_{K})-G_{Na}m^{3}h(V-V_{Na})-G_{L}(V-V_{L}) \tag{1}\]  
\[\frac{dn}{dt}=\alpha_{n}(V)(1-n)-\beta_{n}(V)n \tag{2}\]  
\[\frac{dm}{dt}=\alpha_{m}(V)(1-m)-\beta_{m}(V)m \tag{3}\]  
\[\frac{dh}{dt}=\alpha_{h}(V)(1-h)-\beta_{h}(V)h, \tag{4}\]  
where, \(m\) and \(h\) are the activation and the inactivation variables for the Na\({}^{+}\) channels, respectively; \(n\) is the K\({}^{+}\) inactivation variable; the \(\alpha\) and \(\beta\) rates are functions of V at temperature T = 6.3 \({}^{\circ}\)C [11]. \(n\), \(m\) and \(h\) are dimensionless quantities ranging from 0 to 1. These four partial differential equations quantitatively describe the changes in the membrane as functions of space and time. We use the conductance model to approximate the synaptic activity using the equation [20]:  
\[I=g_{ex}(V-E_{ex})+g_{in}(V-E_{in}), \tag{5}\]  
where \(g_{ex}\) and \(g_{in}\) are, time-dependent excitatory and inhibitory conductances, respectively. \(E_{ex}\) and \(E_{in}\) are the reverse potentials for excitatory and inhibitory synapses, respectively. _Img_ is the normalized intensity of the image in grayscale. The peak conductance is set equal to the image intensity: \(q_{ex}=Img\), \(q_{in}=Img\) and synaptic decay times for excitatory and inhibitory synapses are set to \(\tau_{ex}=4\) ms, \(\tau_{in}=10\) ms, respectively. Figure 1: The electrical equivalent circuit of (**A**) Hodgkin-Huxley neuron model and (**B**) Leaky Integrate-and-Fire neuron model. The time dependency of the conductances is given by two differential equations:  
\[\frac{g_{ex}}{dt}=-\frac{1}{\tau_{ex}}g_{ex}+\sum_{r=-R}^{R}\sum_{f=-R}^{R}\frac{ W_{ex}(r,f)}{A_{ex}}q_{ex}(x-r,y-f) \tag{6}\]  
\[\frac{g_{in}}{dt}=-\frac{1}{\tau_{in}}g_{in}+\sum_{r=-R}^{R}\sum_{f=-R}^{R} \frac{W_{in}(r,f)}{A_{in}}q_{in}(x-r,y-f), \tag{7}\]  
where \(A_{ex}=0.0141\) mV, \(A_{in}=0.0281\) mV. \(W_{ex}(r,f)\) and \(W_{in}(r,f)\) are the Gabor type filters representing synaptic weight matrices with synaptic radius R for excitatory and inhibitory synapses, respectively.