Regulating firing rates in a neural circuit by activating memristive synapse with magnetic coupling

Abstract

The change of firing rates depends on how synaptic features interact with intrinsic properties of cells in neural system. Considering the chemical synaptic features, we design a controllable memristive synapse with magnetic coupling that are voltage-controlled, nonlinear, and unidirectional. To explore the effect of firing rates on interactions between synapse and neuron, the memristive synaptic current involving excitation and inhibition is then mapped into a generalized neuronal model. We observe and characterize the appearance of counterintuitive behavior that increased excitatory memristive synaptic current leads to the decrease in firing rates, and increased inhibitory memristive synaptic current leads to the increase in firing rates in the neural circuit. For the counterintuitive phenomenon, we utilize a geometric dynamics method to provide an underlying dynamics mechanism how the excitatory or inhibitory current impacts the decrease or increase in firing rates.

Appendix: Model description in phasor domain

Here, we design a controllable memristive system composed of the voltage follow, magnetic coupling, and two multipliers. The schematic diagram is shown in Fig. 12.

Fig. 12

The analog circuit composed of two voltage follows (U1, U2), the transformer (T2) to characterize magnetic coupling, and two multipliers (A1, A2). The two red oblique 8-type curves are plotted via analog circuit and numerical stimulation, respectively. Note that the red arrow denotes input voltage and the black arrow denotes output current. (Color figure online)

Note that, our designed memristive system is a voltage-controlled and time-varying devices with only a unidirectional output current $i_{M}$. This system also exhibits current–voltage hysteresis in the inset of Fig. 12. Applying the Kirchhoff’s law into the magnetic coupling part of circuit mentioned above, the coupling equations are obtained by

$$\begin{aligned} \left\{ {\begin{array}{l} i_1 R_1 + j \omega L_1 i_1 - j \omega Mi_2= V_1,\\ - j \omega Mi_1 + j \omega L_2 i_2 - i_2 R_2= V_2. \end{array}} \right. \end{aligned}$$

(S1)

where $i_{1}$, $V_{1}$, and $i_{2}$, $V_{2}$ represent the time-harmonic current and voltage in primary coil and secondary coil, respectively. The voltage follow U1 can guarantee the unidirectional current input, and $V_{1}$ is equal to $V_{\mathrm{pre}}$. R, L, and M are the resistance, self-inductance, and mutual inductance. $j\omega$ denotes the derivative with respect to time variable $\tau$. $\omega$ is the angular frequency. Due to the existence of a voltage follow U2, the current in secondary circuit is zero as $i_{2}= 0$. Thus, Eq. (S1) is rewritten as

$$\begin{aligned} \left\{ {\begin{array}{l} i_1 R_1 + j \omega L_1 i_1 = V_1,\\ - j \omega Mi_1 = V_2. \end{array}} \right. \end{aligned}$$

(S2)

where $R_{1}$ is the controllable resistance to mediate the oscillatory frequency so that there is a different relationship between current and voltage in the whole circuit. Through the analog multiplier A1, we get

$$\begin{aligned} \left\{ {\begin{array}{l} i_1 R_1 + j \omega L_1 i_1 = V_1,\\ V_2^{2} = -\omega ^{2} M^{2}i_1^{2}. \end{array}} \right. \end{aligned}$$

(S3)

The eventual output signal in the phasor-domain representation through the analog multipliers A2 reads

$$\begin{aligned} \left\{ {\begin{array}{l} i_M = \frac{V_2^{2}(V_{\mathrm{post}}-E_{\mathrm{rev}})}{V_0^{2}R_6},\\ j \omega L_1 i_1 = V_1 -i_1 R_1. \end{array}} \right. \end{aligned}$$

(S4)

The eventual output current in the time-domain representation reads

$$\begin{aligned} \left\{ {\begin{array}{l} i_M =\frac{V_2^2 (V_{\mathrm{post}} -E_{\mathrm{rev}} )}{V_0^2 R_6 }, \\ L_1 \frac{\mathrm{d}i_1 }{\mathrm{d}\tau }=V_1 -i_1 R_1. \\ \end{array}} \right. \end{aligned}$$

(S5)

Further, the potential scale transformation is considered as following equations

$$\begin{aligned} \left\{ {\begin{array}{l} x_{\mathrm{pre}} =\frac{V_1 }{V_0 },x_{\mathrm{post}} =\frac{V_{\mathrm{post}} }{V_0 },w=\frac{i_1 }{I_0 },\tau =\frac{t}{L_1 /R_1 }, \\ x_{\mathrm{rev}} =\frac{E_{\mathrm{rev}} }{V_0 },k_1 =\frac{V_0 }{I_0 R_1 },k_M =\frac{\omega ^{2}M^{2}I_0^2 }{V_{0} R_6 }. \\ \end{array}} \right. \end{aligned}$$

(S6)

Substituting Eq. (S6) into Eq. (S5), the memristive synaptic current is given by

$$\begin{aligned} \left\{ {\begin{array}{l} i_M =k_M w^{2}(x_{\mathrm{rev}} -x_{\mathrm{post}} ), \\ \frac{\mathrm{d}w}{\mathrm{d}t}=k_1 x_{\mathrm{pre}} -w. \\ \end{array}} \right. \end{aligned}$$

(S7)

where $i_{M}$ denotes the output current through a memristive system involving magnetic coupling and w is a dimensionless internal state. The approach based on the phasor domain is also able to describe the memristive synaptic current of Eq. (S7) consistent with Eq. (9) obtained from the time domain.