Implementation of Neuro-Memristive Synapse for Long-and Short-Term Bio-Synaptic Plasticity

In this paper, we propose a complex neuro-memristive synapse that exhibits the physiological acts of synaptic potentiation and depression of the human-brain. Specifically, the proposed neuromorphic synapse efficiently imitates the synaptic plasticity, especially long-term potentiation (LTP) and depression (LTD), and short-term facilitation (STF) and depression (STD), phenomena of a biological synapse. Similar to biological synapse, the short- or long-term potentiation (STF and LTP) or depression (STD or LTD) of the memristive synapse are distinguished on the basis of time or repetition of input cycles. The proposed synapse is also designed to exhibit the effect of reuptake and neurotransmitters diffusion processes of a bio-synapse. In addition, it exhibits the distinct bio-realistic attributes, i.e., strong stimulation, exponentially decaying conductance trace of synapse, and voltage dependent synaptic responses, of a neuron. The neuro-memristive synapse is designed in SPICE and its bio-realistic functionalities are demonstrated via various simulations.


Section 1. Memristor Emulator
To design the neuro-memristive synapse, we used the incremental type (i.e., resistance changes from low resistive state to high resistive state) of memristor emulator presented in [28]. The memristor emulator, shown in Figure S1, is designed based on hp TiO2 memristor and exhibits the similar behavioral attributes. Table S1 shows the defining equations of hp TiO2 memristor and the memristor emulator in [28].
However, the ideal equations of TiO2 memristor exhibits linear ionic-dopant drift whereas the fabricated device shows nonlinear ionic-dopant drift features. The Joglekar's window function [29] best fits and perfectly analyzed the nonlinear ionic-dopant drift effects presented at the boundaries (w = 0 and w = D) of TiO2 memristor. Figure S2a shows that the relation between memristance vs. time (i.e., ultimately memristance vs. charge) for the linear and nonlinear models [29] of TiO2 memristors where positive integer p represents the power of Joglekar's window function (FP(x) = 1 −(2x−1) 2P ), where x is state variable. The nonlinear ionic dopant drift of the memristor creates unintended consequences (i.e., erroneous weight programing in neural networks) and to avoid such consequences it is a common practice to initialize the memristor somewhere in the linear region of its memristance range [32,33]. The Joglekar window function, in Figure S2a, shows that even for the highly nonlinear case of p = 1, the memristance vs. time curve exhibits linearity of operation over the range from 2K to 14K. Moreover, both the incremental and decremental memristor emulator itself incorporates the characteristics of Joglekar's nonlinear ionic-drift model as shown in Figure S2b Therefore, we set the initial value of the memristor emulator of the proposed neuro-memristive synapse to M(0) = 2K and limit its operation within the linear region.

Equations of hp TiO2 Memristor Equations of Memristor Emulator [28]
State-dependent Ohm's law: State-dependent Ohm's law: Memristance of incremental memristor emulator: Memristance of decremental memristor emulator:  Figure S3a) and pulse (shown in Figure S3b) inputs, the pinched hysteresis loops pass through the origins. Moreover, the lobe area of the pinched hysteresis loops are decreased with increasing frequency and tends to a singled valued straight line for f ≥ 800 Hz. Therefore, the memristor emulator, with specified intrinsic parameters as a neuro-memristive synapse, exhibits the defining characteristics of a memristor.

Section 2. Voltage Input Response of Neuro-memristor Synapse
To compare the circuit response for current and voltage input, we stimulated the proposed memristive circuit with a voltage input Vin (pulse amplitude PA = 0.75 V, pulse width PW = 1 ms and pulse period PP = 50 ms), as shown in following Figure S4. Observed that the PW and PP of the Vin (in Figure S4a), and the VMFE and VDEP (shown in Figure  S4b) remain same as that of PW and PP of Iin (in Figure 3a), and VMFE and VDEP of Figure 3b of the manuscript. Figure. S4c shows that the current passing through the memristor emulator (Imem) is decreasing with each input pulses. The current Imem (iin in Figure S1) decreases because the potential difference between the nodes of input resistor (Rs) decreases with increasing feedback voltage as shown in Figure S4c. Thus, results in lower current supplies in the capacitor (CT) and resistor (RT) (shown in Figure S1). Therefore, the increments in synaptic strength (Msyn) for later input cycles are smaller than that of initial input cycles as shown in the inset of Figure S4d. Figure S4e shows that the dissimilar increments in Msyn hamper the synaptic voltage accumulations in the later cycle. However, for a current input (iin), the same amplitude and width current is flowing in Rs, and copied to intrinsic capacitor (C) and resistor (R). Therefore, the increment in memristance Msyn (i.e., synaptic strength in Figure 3c of the manuscript) for each cycle is almost similar and results in steady distinguishable increments in synaptic voltage (Vsyn in Figure 3d). The synaptic voltage difference (ΔVsyn = 290 mV) of each cycle is sufficient enough to generate the post synaptic firings like as reference [14].
Due to the dissimilar and slower accumulation of Msyn (in Figure S4d) in later cycle of a given voltage input, the neuro-memristive synapse might exhibits unintended consequences of erroneous weight updating (for spiking neural networks) or synaptic strength modification (for implementation of bio-realistic attributes). Moreover, it is quite difficult to process such small synaptic voltage difference (ΔVsyn ≈ 80 mV) to generate the post synaptic firings like as reference [14]. Therefore, we concluded that the proposed neuro-memristive synapse can operate in voltage mode but it is recommended to operate with current mode to obtain the optimal performance.

Section 3. Shorter and Lengthier Current Input Response of Neuro-memristor Synapse
The pulse duration of biological action potential is typically 1 ms ~ 3 ms. However, most of the neural action potential lasts around 1 ms. Therefore, we chose the pulse width of input stimulation as 1ms.
We stimulated our neuro-memristive synapse with Iin (pulse amplitude PA = 100 µA, pulse width PW = 0.25 ms and pulse period PP = 50 ms) where PW is 4 times shorter than the PW in Figure 3 in the main manuscript, and PA and PP remains same as shown in Figure S5a. Figure S5b shows the same VMFE and VDEP signals as in Figure 3 of the manuscript. As expected, the memristive synaptic build-up, shown in the inset of Figure S5c, for shorter pulse width is lesser than the synaptic build-up in Figure 3 (manuscript). Thus, the memristive synaptic voltage, shown in below Figure S5d, of the proposed synapse is smaller than VSyn in Figure 3, and eventually results in the less likelihood of postsynaptic firings. We further tested our neuro-memristive synapse with Iin (pulse amplitude PA = 100 µA, pulse width PW = 1.5 ms and pulse period PP = 50 ms) where PW is 1.5 times shorter than the PW in Figure 3 (manuscript), and PA and PP remains same, as shown in Figure S6a. Figure S6b shows that the VMFE and VDEP signals are remain same as Figure 3. Expectedly, the memristive synaptic build-up, shown in the inset of Figure S6c, for lengthier pulse width is higher than the synaptic build-up in Figure 3 for which the memristive synaptic voltage, shown in Figure S6c, is higher than VSyn in Figure 3. The higher buildup in synaptic strength and voltage increase the probability of postsynaptic firings.