where ( )ut is the membrane potential, C is the membrane capacitance, R is the input resistance, ()i to current driving the neural state, ()i tj from the j-th synaptic input, and wj ito is the external is the input current represents the strength itj , of the j-th synapse. Both () and () are functions of spike trains, as given in Equation 2. When R " 3 formula 3 is reduced to an IF model. In both IF and LIF models, a neuron is supposed to fire a spike, whenever the membrane potential u reaches a certain value y referred to as the firing threshold. Immediately after the spike, the neuron state is reset to a new value ures 1 y and holds at that level for the time interval representing the refractory period. The majority of the neuromorphic systems utilize IF and LIF neurons as they are easier to implement and are computationally efficient. The LIF model has been extended with one or more adaptation variables to account for different firing patterns. A well-known model is the Izhikevich model, which can produce firing patterns experimentally verified on neocortical and thalamic neurons [26]. However, it is not clear what roles the different firing patterns are playing in learning and cognition, and those additional adaptation variables increase the model complexity. Therefore, they are less used in machine intelligence applications. B. Neuron Dynamics in Spike Response Model (SRM) The aforementioned IF and LIF models are over-simplified by considering the synaptic connection as a time-invariant device with a constant efficacy w and assume that the membrane potential reset as an instantaneous procedure. A more realistic neuron model considers the dynamics in the neuron and synapse behavior. The arrival of a presynaptic spike triggers the synaptic electric current flowing into the biological neuron [20]. It causes a change in the membrane potential of the synapse, which is referred to as post-synaptic potential (PSP). In a general form, the time course of jth PSP can be described as the convolution of the presynaptic spike train () Stj and a kernel function () scaled by a weightKtj coefficient wj PSPjj jj as the following: () tw St sK ss wK tt ==jj tt / # jl 1 where tj l is the time of spikes on the input ().Stj K(t) can be an exponential, a dual exponential or an alpha kernel defined by following equations: Kt e () = () t = x - Kt t e- t x x Kt Ve e SECOND QUARTER 2022 () ()0 ms tt =-xx -(5) (6) (7) 3 () () (), jl d (4) 1 0.8 0.6 0.4 0.2 Their spike responses are illustrated in Figure 2. The reset of the membrane potential is no longer instantaneous. Instead, it is modeled as a negative potential induced by the output spike train () Sto kernel function h(t), () () () Rt =- =- o #3 St sh ss ht tl(), ttlo1 o d / going through a (8) where to l is the time of spikes on the output spike train. Usually h(t) is a kernel given in Equation 5. The way to interpret the neuron dynamics is as a convolution of the impulse response of a filter with the input spike train as in Equations 4 and 8, and is referred to as the Spike Response Model (SRM). The membrane potential is the combined effect of PSP(t) and R(t) as shown in Figure 3. Using SRM representation, it can be represented as an integral over the past input and kernel responses. A typical SRM model is defined as the following [12]: N Vt wK tt Vh tto = jo1 ll mj j where () Vtm =- -1 () () t j th / () // ll (9) tt is the membrane potential, K(t) and h(t) are two convolution kernels associated to synaptic dynamics and membrane potential reset events. When () Vtm exceeds the threshold Vth, the neuron generates a spike output whose time is indicated by the spike time t .o l By using a kernel K(t) with arbitrary shape, the SRM model provides complicated dynamics and rich temporal information. The simplified LIF model in Equation 3 is a special case of the SRM model, where the K(t) and h(t) are two low-pass filters. The SRM model shows that the membrane potential is a function based on not only the current but also the past input spikes, which explains the neuron's ability to respond to temporal patterns. The kernels in the SRM model can be implemented as discretized digital filters. Using the Z-transform [30], [31], they can be represented as a Linear Constant-Coefficient Difference (LCCD) equation in the following form: P yt =- +== 11 ab pyt pxtp q [] Q 6 @ // (10) p q [], Exponential Kernel Alpha Exponential Kernel Dual Exponential Kernel 020406080 100 Time Figure 2. Exponential, Alpha and dual exponential Kernels. IEEE CIRCUITS AND SYSTEMS MAGAZINE 9 PSP

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