2.3. Switched-capacitor implementation of a bistable stochastic synapse

2.3.1. Model
The stop learning model of long-term plasticity has been introduced in Brader et al. (2007), based on earlier work in Fusi et al. (2000). The model represents a synapse with two stable states, potentiated and depressed, whereby the state transition between both stable states is regulated via a continuous internal state X(t) of the synapse. X(t) is influenced by a combination of pre- and postsynaptic activity, namely the presynaptic spike time tpre and the value of the neuron membrane voltage Vmem(t). A presynaptic spike arriving at tpre reads the instantaneous values Vmem(tpre) and C(tpre). The conditions for a change in X depend on these instantaneous values in the following way:
(5)  X → X + a   i f   { V m e m   ( t p r e   ) > θ V    a n d                                                     θ u p  l  < C ( t p r e   ) < θ u p  h  }      (6)  X → X − b   i f   { V m e m   ( t p r e   ) ≤ θ V    a n d                                                     θ d o w n  l  < C ( t p r e   ) < θ d o w n  h  } ,      where a and b are jump sizes and θV is a voltage threshold. In other words, X(t) is increased if Vmem(t) is elevated (above θV) when the presynaptic spike arrives and decreased when Vmem(t) is low at time tpre. The θlup, θhup, θldown, and θhdown are thresholds on the calcium variable. The calcium variable C(t) is an auxiliary variable (see Brader et al., 2007 for details) that provides a low-pass filter of the postsynaptic spikes. This gives the ability to stop the learning based on thresholded, long-term averages of postsynaptic activity. In the absence of a presynaptic spike or if stop learning is active [i.e., C(t) hits the respective threshold], then X(t) drifts toward one of two stable values:
(7)  d X  d t   = α        i f   X > θ X        (8)  d X  d t   = − β   i f   X ≤ θ X         The bistable state of the synapse is determined according to whether X(t) lies above or below the threshold θX. Computationally, this model is interesting because through X(t) it can learn a graded response to an input pattern even though the output weight of the synapses is binary. The model also has some biological veracity, being sensitive to pre-post and post-pre spike patterns in a manner similar to the well-known spike time dependent plasticity (Brader et al., 2007).

2.3.2. Circuit implementation
The circuit schematic shown in Figure 7 replicates the model described in Equations (5–8). In contrast to the circuit presented in (Indiveri et al., 2006) our implementation makes use of SC technique. Thus, the model equations are solved in a time-discrete fashion, which enables the use of low-leakage switches as shown in Section 2.2.3 to achieve very low drift rates α and β. The time-discretization also allows for time multiplexing the single synapse circuits, thus, one driver circuit (see blue box in Figure 7) can drive multiple (in our case 64) synapses (red boxes). Due to the removal of active elements, one synapse circuit can be reduced to only 2 capacitors and 4 low-leakage switches storing the synapse state X (cp. Equations 5–8) as a differential voltage. The synapse occupies an area of 3.6 μm × 3.6 μm which is shared equally by the two synapse capacitors with 22 fF each. These are custom-designed metal-oxide-metal capacitors, utilizing an interdigitated fingered layout in the complete 5-layer metal stack with cut-outs on the lower two layers for wiring. The low-leakage switches are located directly below the capacitors. Each synapse can be connected to the driver circuit via switches Ssyn,i, where i indicates the column number in the synapse matrix, and 4 wires VINP,VINN,VXP, and VXN. The driver circuit is basically an SC integrator, which integrates different voltages Vα, Vβ, Va, and Vb in dependence of synapse state, neuron state and incoming presynaptic spikes onto the synapse capacitors Csyn,i. The integrator's opamp is the same as for the presynaptic driver presented in Section 2.2.4. As shown in the timing diagram in the lower right corner of Figure 7, the operation principle can be divided into 4 phases “Reset,” “Readout,” “Comparison” and “Integration” for one synapse. All synapses of one row are cycled through sequentially, whereas all rows are processed in parallel.
Figure 7  LTP circuit. In the reset phase an offset compensation of the opamp (cp. Section 2.2.5) is performed, which avoids the integration of a possible offset voltage as well as residual charge on the relatively long wires to the synapses. Therefore, switches annotated with Φreset are closed, which closes a negative unity-gain feedback loop around the opamp. The offset voltage appearing at the opamp input is then stored on capacitors Crefr and Chebb and can be subtracted in the integration phase.
After reset a readout of the synapse state is performed. Switches Ssyn,i of the currently active synapse i are closed, which places the synapse capacitors in the feedback path of the opamp. The voltage stored on the capacitors, i.e., the synapse state X, is now visible at the opamp output between the differential lines VXP and VXN.
When the readout is completed the synapse capacitors stay connected and a comparison of the synapse state with threshold ΘX is performed. In the implementation ΘX is fixed at 0.5, thus, the comparator (see Section 2.3.3) only has to compare whether VXP > VXN. After comparison the result is provided by signals comp and its inverted counterpart comp_n.
In the integration phase the refresh part (see Equations 7, 8) and the hebbian part (Equations 5, 6) of the learning model are performed. In this phase switches annotated with Φintegrate are closed. If comp is high then the differential synapse voltage VX is increased by CrefrCsyn·(Vα−Vcm), otherwise it is decreased by CrefrCsyn·(Vβ−Vcm). This results in refresh rates of
(9)  α = C r e f r    C s y n     · ( V α  − V c m   )  Δ t      and
(10)  β = C r e f r    C s y n     · ( V β  − V c m   )  Δ t   ,    where Δt = 0.62 ms, which is the time needed for processing the 64 synapses of a row sequentially (in biological real-time mode).
If a presynaptic input spike arrives, then switch signal pre is high during the integration phase. In dependence of the postsynaptic membrane state ΘV signals learn_up and learn_down are set. The neuron circuit providing the membrane state is an SC leaky integrate-and-fire neuron presented in the companion paper Mayr et al. (in press). It is equipped with two comparator circuits for spiking threshold detection and for judging the current membrane state, i.e., the Vmem(tpre) ≷ θV condition of Equation (5) resp. Equation (6). If Vmem(tpre) > θV, then learn_up is high and learn_down is low (neglecting the “stop learning” mechanism for now). Thus, the upward jump size is calculated by
(11)  a = C h e b b    C s y n     · ( V a  − V c m   ) .    If Vmem(tpre) < θV, then learn_up is low and learn_down is high, which results in the downward jump size of
(12)  b = C h e b b    C s y n     · ( V b  − V c m   ) .    In order to reduce the number of control voltages, single-ended input voltages are provided. The resulting common mode offset, caused by this asymmetry, is compensated by the SC CMFB circuit.
The “stop learning” feature described in Section 2.3.1 is handled by setting learn_up resp. learn_down to low using combinational logic (not shown). Therefore, the state of the calcium variable can be calculated externally in an FPGA, where the postsynaptic spike train is filtered by a low pass filter. The low pass filter output is then compared against the stop learning thresholds θlup, θhup, θldown, and θhdown and the two resulting binary signals for enabling learning in the up and down direction, respectively, are transmitted to the driver circuit. As an additional feature for testing we implemented a “learn force” mode where learn_up and learn_down can be set explicitly, similar to keeping the neuron membrane permanently elevated or depressed.
The comp signal, which is provided in the “Comparison” phase states whether the synapse is depressed (LTD) or potentiated (LTP). This binary output is used to scale the PSC generated by the presynaptic adaptation circuit (see “Weight Scaling and Charge Transmission” in Figure 3). Therefore, each synapse has two 4-bit weights for LTP and LTD stored in a RAM (see Figure 1), which is chosen accordingly to the synapse state and transmitted to the weight scaling circuit. The scaling of the PSC is done via binary weighted capacitors, transferring charge to the neuron circuit. Additionally each synapse is selectable excitatory or inhibitory, which inverts the PSC voltage. Thus, inhibitory stop-learning synapses are also possible.

2.3.3. Comparator circuit
A circuit schematic of the comparator shown in Figure 7 is depicted in Figure 8A. It consists of a preamplifier (see Figure 8B), which is inspired by Dessouky and Kaiser (2000) and a simple dynamic latch circuit (Song et al., 1995) shown in Figure 8C. This architecture has been chosen, because the dynamic latch circuit can have a high random offset voltage of up to 20 mV, caused by mismatch. The preamplifier raises the differential signal level to minimize decision errors, caused by this mismatch. The preamplifier is therefore equipped with an offset compensation (compare Section 2.2.5). At the output of the comparator circuit an SR-latch is connected, which stores the result until the next comparison.
Figure 8  (A) Comparator circuit with offset-compensated preamplifier, compensation capacitors Cc and latch circuitry. (B) Preamplifier circuit schematic. (C) Latch circuit schematic.