The dynamics of the NMDAR gating variable is characterized by a fast rise followed by a slow decay. When the presynaptic inputs at a recurrent synapse in a neural population are described by a Poisson spike train at a rate of r per cell, the slow dynamics of the average NMDAR gating variable S is characterized by the non-linear differential equation as follows (Wong and Wang, 2006):

where τ is the decay time constant, and γ is a gain coefficient. To implement the circuit of the NMDAR gating variable, the corresponding mathematical expression is approximately transformed into the following current-mode description:

where dimensionless variables S and γ are replaced by two ratios of currents ISIref and IγIref, respectively. Because of the current-mode design, we use a current I_r to represent “the firing rate” r in this work, and we call I_r the neural activity in the remainder of this paper. The time constant τ=CUTIτ.

We used a dynamic voltage-current circuit (DVI) originally proposed for the log-domain circuit (Yu and Cauwenberghs, 2010) to achieve the derivative on the left side of Equation (2) (Figure 2A). Since

the dynamics of the voltage difference V_S − V₀ has the following relationship with I_S/I_ref:

By combining Equation (4) and Equation (2), we obtain the following voltage-mode differential equation to characterize the dynamics of S:

According to the dynamical systems approach of circuit synthesis (Arthur and Boahen, 2011), three currents are required to drive the capacitor in the dynamic voltage-current circuit (Figures 2A, 3A): two currents decrease the capacitor voltage and one increases it. The first current term I_τ corresponds to the time constant of the circuit. The smaller its value is, the longer the time constant. The second term equals the product of the gain current I_γ and the activity current I_r, divided by the gating variable current I_S. This current is generated by a multiplier (Figures 2, 3) on the basis of the translinear principle for subthreshold transistors (Gilbert, 1975; Papadimitriou et al., 2013). It indicates that the driving force toward V_DD decreases with increasing I_S. The third term corresponds to the driving force toward ground, and can also be generated by a multiplier (Figure 3A). Because both the second and third terms have the product of I_γ and I_r, their multiplier circuits can share a common left branch of the multiplier in Figure 2B.

We concentrated on the steady state and the time constant for a given configuration as major dynamic characteristics of this circuit. From Equation (2), we obtain the steady state S¯ and the time constant τ as follows:

Circuit simulations in Cadence in our previous work (You and Wang, 2016) have demonstrated that the NMDAR gating variable circuit can implement the major dynamics of its biological counterpart characterized by Equation (6): S¯ increases with increasing I_r and I_γ but decreases with increasing I_τ; the time constant of the circuit τ decreases with increasing I_τ but is independent of I_γ and I_r.

For the sake of neuromorphic implementations, the original reduced two-variable decision model (Wong and Wang, 2006) is re-expressed in the current-mode description as follows:

where i (= 1, 2) labels two selective populations. For simplicity, we adopted the linear-threshold unit (LTU) to model the neural activities of two populations I_{r, i}, which are given by the equation as follows:

where the dimensionless variable IgainrIrefr is the gain of the neural population, and IrefrIgainrIthrr is the threshold current. I_{syn, i} denotes the corresponding synaptic current. [x]⁺ is equivalent to the function max(x, 0). According to Equation (8), we require only a multiplier and several current mirrors to realize the LTU circuit (Figure 3C). However, the neural activity I_r of this LTU circuit in circuit simulations increases smoothly, not suddenly, as described by Equation (8) when the synaptic current I_syn approaches the threshold current. Therefore, we chose a new activation model instead of Equation (8) in the following system approach analysis:

where the parameter g tunes the smoothness of the activation curve around the threshold current. Equation (9) is more biophysically realistic for the response of neural populations.

The total synaptic currents of two selective populations I_{syn, i} are given by the following equations:

where I_{sti, i} represents the external sensory stimulus to the population i, and I₀ is the effective background input. Iw+ and Iw- are the effective self-excitation and mutual-inhibition currents per unit (the gating variable) in the recurrent network, respectively. The synaptic current block can also be realized by two multipliers and several current mirrors (Figure 3B). However, for this synaptic current circuit shown in Figure 3B, I_syn is close to but still larger than 0 when I_s1 < I_s2, indicating that our synaptic current circuit looks like a rectifier ([·]⁺). Although this result is not consistent with the response determined by Equation (10), it has little influence on the implementation of attractor dynamics in the decision circuit, owing to the rectifying effect of the LTU circuit.

We built the neuromorphic circuit of this two-variable WTA model on the basis of Equation (7), (8), and (10). Figure 1B shows the block representation of the WTA circuit.

Configuring parameters of the simplified two-variable WTA model described by Equations (7–10) allows implementation of decision-making, working memory and hysteresis in visual perception (Wong and Wang, 2006; You et al., 2011). In this work, to test the responses of the neuromorphic WTA circuit built in Cadence (Figure 1), we configured the circuit and design simulation protocols for three cognitive tasks. Bias currents configured in the circuit are as follows: I_τ = 5pA, I_γ = 10pA, I_ref = 100pA, Igainr=100pA, Ithrr=50pA, Irefr=100pA, I₀ = 15pA. The capacitance of the capacitors used in the circuit of the NMDAR gating variable C = 20pF. The parameter g = 0.2 from simulations. We also used the above values as parameters in the model for theoretical analysis. Furthermore, U_T = 25mV in the model. Bias currents Iw+ and Iw- are task specific.

We tested the responses of this decision circuit through simulations given the stimulus with different coherence levels, which were given after 0s (Figure 5A). Responses of both populations stay at a low activity before the stimulus onset. After the stimulus was presented, one of them exhibited a gradual ramp-up activity, and the other exhibited a gradually decreasing activity. According to the decision bound theory supported by some observations in the monkey experiment (Kiani et al., 2008), we set a fixed neural activity (corresponding to a fixed I_r 50pA) as the decision threshold in our circuit simulations. A decision is made once the activity of any population exceeds the decision threshold, and corresponding time is measured as the reaction time. The speed of the ramp-up activity increased with increasing coherence of the stimulus, whereas the reaction time decreased (Figure 5B); both these observations are consistent with neural activities and behaviors observed with the monkey experiments (Shadlen and Newsome, 2001). However, because our decision circuit is a noise-free system, the reaction time had a linear relationship with the logarithm of the coherence level (Figure 5B), as has been observed in previous studies (You, 2014).

These simulation results are consistent with results from a phase-plane analysis of the model described by Equation (7), (8), and (10) (Figure 5C). In the (S₁,S₂) phase space (decision space), two lines called nullclines are plotted first by setting the dynamical equations dS₁/dt = 0 and dS₂/dt = 0 (Equation 7). The intersections of two nullclines are steady states of the model. The direction field in the phase space shows how the system state will evolve. In the absence of a stimulus, the two nullclines intersected with each other only once, thus producing a stable steady state, that is, they function as an attractor (Figure 5c1), thus explaining why the two populations in the decision circuit had low spontaneous activities before the stimulus onset. When an unbiased stimulus was presented (Coh. = 0%), two nullclines intersected with each other three times and produced a saddle-type unstable and two stable steady states (Figure 5c2). They formed two basins of attraction around different two attractors, representing two alternatives in the decision. The system state was located in the vicinity of the unstable steady state before the stimulus onset and then evolved toward one of two attractors after the stimulus onset. This saddle node structure determined the time course of neural activities underlying the two-alternative decision. When a biased stimulus was applied, the phase space was no longer symmetrical. As an example of the biased stimulus Coh. = 12.8% (Figure 5c3), the unstable steady state was closer to the attractor representing the second alternative, thus indicating that the attractor state responsible for the first alternative has a larger basin of attraction than the other. After the onset of a biased stimulus, the initial state of the system already lies within the basin of the attractor of the first alternative, and the system state will evolve toward its favored attractor, especially in our noise-free circuit.

Figure 6A shows the bifurcation diagram of a neural population with the unbiased external stimulus I_sti as a parameter in the WTA model for the decision task and presents the relationship between the steady states of the system and the stimulus strength. In this bifurcation diagram, the gradually increasing middle curve represents symmetric steady states, and the upper and lower curves represent asymmetric states. Blue solid curves and red dashed curves denote stable and unstable steady states, respectively. The neuromorphic WTA circuit shows spontaneous activities at a low rate when the stimulus is small (Figure 6CI) in the decision task, because there is only one stable steady state for the weak stimulus (the region I in Figure 6A). As the external stimulus increases further, the system has three stable steady states (two asymmetric and one symmetric) and two unstable steady states (asymmetric). In this case, the system shows stronger spontaneous activities. For the stronger external stimulus, however, the system will develop one unstable symmetric steady state and two stable asymmetric steady states in the phase space (regions II–IV in Figure 6A), thus leading to WTA behavior (Figures 6CII–IV). The time constant of the unstable manifold of this symmetric state is very large and decreases with increasing stimulus strength for low strength in this region, but increases again for high strength (Figure 6B). The simulation results of the circuit were also consistent with the prediction of its bifurcation analysis for the WTA model. The divergence speed of neural activities for the stimulus strength around regions II and IV is larger than that around region III (Figures 6CII–IV). When the stimulus strength further increases, and the symmetric steady state becomes stable again, the system has another two asymmetric stable steady states and two asymmetric unstable steady states (Figure 6A). If the stimulus strength is too large, the system has the sole steady state that is stable. The system cannot develop the ramping-up activity and make a decision (Figure 6CV).

We concentrated on following aspects of working memory when we performed this cognitive task in the neuromorphic WTA circuit: the ability of working memory and the limitation of its capacity. In circuit simulations, we verified the working memory capacity limit under two circuit configurations (Iw+=240pA and 300pA). We also applied the stimuli with one or two items to the circuit during the cue presentation according to the protocol of working memory (Figure 4B).

The neural activity patterns of the circuit with Iw+=240pA in the circuit simulations are shown in Figure 7A. Several characteristics are worth noting. First, two neural populations were spontaneously active at a low rate before the onset of the cue stimuli. Second, the neural populations, which encoded the directions of the items that appeared during the cue period, increased their neural activities. When only one item was presented, only the corresponding neural population fired (Figure 7a1). When two items appeared, both neural populations developed strong activities (Figure 7a2). Third, the strong activity persisted throughout the delay period when the working memory load comprised one item (Figure 7a1), but the strong activities decayed gradually when the load size was two (Figure 7a2). The results from the above circuit simulations revealed that the WTA circuit with Iw+=240pA could remember only one item.

The results from the phase-plane analysis for the WTA model were also able to account for these observations (Figure 7C). For the system without the cue presentation (Figure 7c1), two nullclines intersected with each other five times, thus producing three stable steady states and two unstable steady states. These three stable steady states, around which three basins of attraction formed, were divided into two classes: one symmetric low state, called the spontaneous state, and two asymmetric high states, called memory states. The WTA circuit stayed at this spontaneous state before the cue presentation. After the onset of the cue, if the memory load was one item, the system had only one attractor (Figure 7c2), and then one population in the WTA circuit evolved from its spontaneous state to the high activity (Figure 7a1). If the memory load was two items, the system had three attractors (Figure 7c3). Because the initial state of the circuit was located in the attraction basin of the attractor with two high values, the activities of both populations increased (Figure 7a2). After the cancellation of the cue, the system evolved from the attractor in Figure 7c2 to its closest memory state (the south-east asymmetric attractor) in Figure 7c1 for one item, but from the attractor with two high values in Figure 7c3 to the spontaneous state in Figure 7c1 for two items. These results illustrate why the WTA circuit could hold only one item in its working memory.

When Iw+ was configured to be 300pA, all populations subjected to the cue items, showed persistent activities after the cue stimuli offset (Figure 7B). The memory capacity was two in this configuration, which was also illustrated by the similar phase-plane analysis for the WTA model. Given the cue stimuli with one item and two items, the dynamics of the system characterized by Figures 7d2,d3 was similar to those of Figures 7c2,c3. Therefore, the WTA circuit behaved similarly during the cue presentation for both configurations of recurrent excitatory connection strength (Figures 7A,B). However, two nullclines intersected nine times, producing four basins of attraction, among which there was a memory attractor with two high values representing the memory of two items (Figure 7d1). In the case of the stimuli with two items, the system state did not evolve toward the spontaneous state but instead evolved toward its closest memory state responsible for the memory of two items. Thus, the WTA circuit was able to hold persistent activities during the delay and remember two items (Figure 7b2).

Our previous work has revealed that hysteresis in visual perception can be reproduced in the simulation of the two-variable WTA model (You et al., 2011). In circuit simulations, we applied the stimulus protocol of the discrimination task between the man's face and the kneeling girl to the neuromorphic WTA circuit with three different Iw+. Corresponding activity plots are shown in Figure 8A. In the circuit simulation for Iw+=100pA (Figure 8a1), the activity of the population representing the man's face (solid curve) gradually decreased and that of the kneeling girl (dashed curve) increases with the changing stimulus. Notably, the crossover of two activities (15.5s) was followed by the crossover of the stimulus to two populations (10s). Thus, the circuit perceives the kneeling girl later than the time at which its feature exceeds the feature of the man's face; that is, a hysteresis phenomenon in the visual perception was observed. The corresponding hysteresis loop of neural activity is shown in Figure 8. Similarly, a hysteresis phenomenon was also observed for Iw+=80pA, but the corresponding hysteresis effect weakened (Figures 8a2,b2). If Iw+ decreased further, for example, 60pA, the hysteresis effect disappeared (Figures 8a3,b3).

To compare the circuit simulation results and theoretical analysis results, we calculated the steady states of the WTA model with the variant parameter Iw+ and the increasing feature coherence Coh. However, because of the unprecise mapping of the WTA model onto the neuromorphic WTA circuit, we chose another set of Iw+s (120, 100, and 80pA) in the WTA model for comparisons. When Iw+=120pA, the system had strong self-excitation. Along with the increase in Coh, the number of steady states of the system changed from one to three and then to one (Figure 8c1). There was a Coh interval around 0, in which the system had three steady states, including two stable and one unstable. These steady states accounted for the hysteresis in the visual perception (Figures 8a1,b1). If Iw+ decreased and was equal to 100pA, the Coh interval with three steady states decreases (Figure 8c2), thus revealing a weakened hysteresis effect of the visual perception (Figures 8a2,b2). When Iw+=80pA, the Coh interval with three steady states disappeared and the system had a single steady state (Figure 8c3). Thus, there was no hysteresis in the visual perception (Figures 8a3,b3). Although there was a slight mapping mismatch from the WTA model onto the neuromorphic WTA circuit, the above circuit simulation results and theoretical analysis results demonstrated that the neuromorphic WTA circuit mimics hysteresis behavior in the visual discrimination task.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.