Abstract
We propose a new digitaltoanalog converter (DAC) for realizing a synapse circuit in mixedsignal spiking neural networks. We refer to this circuit as a “timedomain DAC (TDAC)”. It produces weights for converting a digital input code into voltage using one current waveform. Therefore, the TDAC is more compact than a conventional DAC consisting of many current sources and resistors. Moreover, a TDAC with leak resistance reproduces biologically plausible synaptic responses expressed as alpha functions or dual exponential equations. We also present numerical analysis results for a TDAC and circuit simulation results for a circuit designed using the TSMC 40 nm CMOS process.
1 Introduction
Applicationspecific integrated circuits (ASICs) for neuromorphic hardware have been studied intensively with the aim of achieving highly efficient computation [2, 5, 11, 18, 20, 25]. Although most neuromorphic hardware is designed with digital circuits [2, 4, 8,9,10, 12, 22], some studies have sought higher efficiency by using analog circuits [11, 18] owing to the advantage that highly efficient multiplyaccumulate (MAC) operations can be achieved.
Neuromorphic computation requires many MAC operations for the input signals and synaptic weights, and therefore, implementation of highly efficient MAC operations is important for realizing highly efficient neuromorphic hardware. To improve the energy efficiency of MAC operations, attempts have been made to (i) binarize the synaptic weights, (ii) realize weighted summation using current (wiredsum), (iii) realize weighted summation using capacitors, and (iv) develop a method that combines (i) and (ii) or (i) and (iii).
Bankman et al. [5] realized a highly energyefficient binary convolution neural network chip as a recognition processor. In the circuit, the synaptic weights express binary information and are stored in digital memory. The weighted summation is then calculated by converting the binary information into analog voltages. The capacitors on this chip for weighted summation work also as a digitaltoanalog converter (DAC), and this circuit is categorized as a mixedsignal circuit. A mixed–signal circuit that has multibit synaptic weights requires a large footprint area and a large amount of energy for the DAC, but the chip developed by Bankman et al. has low power consumption because the synaptic weights are restricted to be binary and the weighted summation is realized by capacitance coupling.
Weighted summation by current is often used in analog neuromorphic hardware [11, 15, 21, 25, 28]. In such circuits, the current is produced by synapse circuits. Currentbased summation is implemented very simply by connecting each of the metal lines of the synaptic output. In analog neuromorphic hardware, synaptic weights are often expressed by analog voltages that are maintained by capacitors. A circuit with this architecture is highly efficient, but it is difficult to reuse the capacitors that hold the synaptic weights because they cannot hold the weights for a long time due to charge leakage. To solve this problem, there have been many studies of synapse circuits that use analog memory [1, 6, 13, 23, 26]. However, many problems need to be solved in order to establish analog memory as a reliable technology. For example, ferroelectric random access memory (FeRAM) suffers from a parasitic charge trapping effect that degrades endurance [29]. Resistive random access memory (ReRAM) is expected as highintegration and highspeed memory, but it may exhibit stochastic switching behavior depending on the composition of the filaments, and this mechanism is still not well understood [7].
To realize highly efficient neuromorphic hardware with reusable synaptic weights that is fabricated using conventional complementary metal–oxide–semiconductor (CMOS) technology, synaptic weights of the circuit architecture can be stored in digital memory and MAC operations can be achieved by an analog circuit; in other words, a mixedsignal architecture is suitable. Realizing a mixedsignal circuit that has multibit synaptic weights is important for achieving onchip learning, but it is difficult to realize highintegration and highly efficient neuromorphic hardware because conventional multibit DACs comprise many currentsource circuits or resistor arrays, thereby necessitating a large footprint and high power consumption [14, 17]. It is especially difficult to implement a highly energyefficient asynchronous spiking neural network (SNN) chip that has multibit synaptic weights because using one DAC per time division is difficult in an asynchronous system.
Figure 1 shows the principal circuits of conventional 4bit DACs where \(S_{0}\)–\(S_{3}\) are the individual values of a bit code, \(V_{ref}\) is the reference voltage, \(I_{ref}\) is the reference current, \(V_{out}\) is the output voltage, and \(I_{out}\) is the output current. Most conventional DACs consist of circuit elements (CEs) for realizing the weight of the bit code (e.g., resistors, capacitors, and current sources) and switches. It is difficult to implement the resistor string type (Fig. 1(a)), the current array type (Fig. 1(b)), and the capacitor array type [3] (Fig. 1(c)) in highly integrated forms because in those cases the number of CEs increases exponentially as the DAC bit length n increases. However, the R2R ladder type [24] (Fig. 1(d)) and the MOS ladder type (Fig. 1(e)) are highly advantageous because the number of CEs increases only linearly with n. Based on Fig. 1, the CE components and their numbers are summarized in Table 1. Our work will be described later.
In general, resistors and capacitors necessitate a larger footprint than do transistors in CMOS technology, and the CEs are designed to have larger dimensions than transistors that consist of logic gates in order to reduce the influence of fabrication mismatch. Therefore, to implement highintegration and highly efficient mixedsignal SNN hardware with onchip learning having multibit synaptic weights, a suitable DAC architecture is one with neither resistors nor capacitors, or one but with few CEs, the number of which does not increase with the number of bits.
Toward SNN hardware, we propose a new DAC circuit that weights each bit of digital memory by using a current (or voltage) waveform. This makes our DAC more compact than conventional ones, and we refer to this DAC as a “timedomain” DAC (TDAC). A TDAC has the advantages that (i) it can reproduce a postsynaptic potential (PSP) and (ii) its number of CEs is independent of the bit width. Herein, we present the results of numerical analysis of TDAC and circuit simulation.
This paper is organized as follows. In Sect. 2, we explain the principle of TDAC, and in Sect. 3, we present a numerical analysis. Circuit design of TDAC and circuit simulation results are shown in Sect. 4. Finally, we conclude the paper in Sect. 6.
2 Circuit Principle of TimeDomain DigitaltoAnalog Converter
Figure 2a shows the TDAC principle in circuit form, and we explain the operation of a 4bit TDAC as an example. The TDAC consists of an analog block and a digital block that comprise AND gates, an OR gate, switches, a switched current source (SCS), resistors, and capacitors. The digital memory values are \(B_1\)–\(B_4\), and in this example, \(B_1\) and \(B_4\) are the least significant bit (LSB) and the most significant bit (MSB), respectively. \(S_{in}(t)\) is the trigger signal for activating the DAC and corresponds to spike pulses when the TDAC is used as the output stage of a synapse circuit. Signals \(S_{B,1}(t)\)–\(S_{B,4}(t)\) are nonoverlapping digital signals with pulse width \(t_w\) for the DAC, and these values are either zero or unity. The SCS outputs a current \(I_{out}(t)\) (\(\propto V_{non}(t)\)), and the leak resistance \(R_{out}\) is an option for realizing the waveform of synaptic potential.
The DAC process using the TDAC without the leak resistance \(R_{out}\) is as follows (Fig. 2b):

(1)
When \(S_{in}(t)\) is high, \(V_{non}(t)\) is set to \(\mathrm {V_{set}}\).

(2)
When \(S_{in}(t)\) is turns off, \(S_{B,4}(t)\) is generated at the trailing edge of \(S_{in}(t)\). At the same time, \(V_{non}(t)\) increases exponentially with time constant \(C_{lk}R_{lk}\). If \(B_{4}\) is high, then capacitor \(C_{out}\) is charged during \(t_w\) by a current proportional to \(V_{non}(t)\).

(3)
\(S_{B,3}(t)\) is generated at the trailing edge of \(S_{B,4}\). If \(B_{3}\) is high, capacitor \(C_{out}\) is charged during \(t_w\) by a current that is proportional to \(V_{non}(t)\).

(4)
Operation 3) is repeated until \(S_{B,1}(t)\) is generated.
For example, \(V_{out,1110}\) represents the voltageconverted digital input code \((1110)_2\) (Fig. 2b) that is obtained by charging, charging, charging, and not charging.
Conventional DACs use many resistors or current sources to weight each bit of digital memory. Implementing these circuit components causes the footprint area to be squared for every unit increase in the length of the memory. By contrast, a TDAC uses a current waveform to weight each memory bit. In the TDAC, the numbers of AND gates, OR gates, and signal generators of \(S_{B,k}\) needed to sample the current waveform increases as the length of the memory increases. Therefore, the number of transistors increases linearly, but the disadvantage is that the time taken by the DAC increases. A TDAC is suitable for hardware that must be compact and operate at low speed, such as SNN hardware. In this example, the output current is positive but the TDAC can output negative current.
3 Numerical Analysis of TimeDomain DigitaltoAnalog Converter
3.1 TimeDomain DigitaltoAnalog Conversion Without Leak Resistance
We define the time of the trailing edge of \(S_{in}(t)\) to be zero, and we express the voltage \(V_{out}(t)\) as
where q is the memory length, t is continuous time (\(t \ge 0\)), and \(f_{scs}(\cdot )\) characterizes the SCS. In this section, we analyze the TDAC when \(V_{non}(t)\) is described by \(V_{set}\exp (\frac{t}{C_{lk}R_{lk}})\) and \(f_{scs}(V_{non}(t))=V_{non}(t)\). Under these conditions, Eq. (1) is expressed as
We integrate Eq. (2) to obtain
To ensure that the DAC characteristics remain linear, the weight of the upper bit of the adjacent must be twice that of the lower bit, namely
We solve Eq. (4) to obtain
Figure 3 shows the input–output characteristics of the TDAC obtained from numerical simulation for varying \(\frac{t_w}{C_{lk}R_{lk}}\). As shown therein, the characteristics are linear when \(\frac{t_w}{C_{lk}R_{lk}} = \ln 2\) but nonlinear otherwise. In particular, monotonicity is also lost when \(\frac{t_w}{ C_{lk}R_{lk} } < \ln 2\).
3.2 TimeDomain DigitaltoAnalog Conversion With Leak Resistance
When a TDAC is used as the output stage of a synaptic circuit, we can view \(C_{out}\) as the membrane capacitance of an analog neuron circuit, where the leak resistance \(R_{out}\) is connected in parallel with \(C_{out}\). In this situation, the temporal variation of \(V_{out}(t)\) with \(R_{out}\) is expressed by
We will show that Eq. (6) fits well with biological data on synaptic potentials when all the memory bits are unity and \(qt_w\) is sufficient large, in which case \(\sum ^{q1}_{k=0} S_{B,qk}(t)B_{qk}\) is nearly unity regardless of t. We assume that \(V_{out}(0)\) is zero, and in this case Eq. (6) is expressed as
We solve Eq. (7) by using the method of variation of constants, setting \(C_{out}R_{out}=\tau _1\) and \(C_{lk}R_{lk}=\tau _2\). The solution of Eq. (7) is given by
where \(g(\cdot )\) is a function of time. By using Eqs. (7) and (8), \(\frac{dV_{out}(t)}{dt}\) is expressed as
If \(\tau _1=\tau _2\), then the integral of Eq. (9) with respect to t is \(tV_{set}\) because \(g'(t)\) is a constant. In this case, \(V_{out}(t)\) is given by
which is an alpha function.
If \(\tau _1\ne \tau _2\), then the integral of Eq. (9) with respect to t is
where \(g_c\) is a constant of integration. By using \(V_{out}(0)=0\), we obtain \(g_c = \frac{\tau _1\tau _2}{\tau _1\tau _2}V_{set}\). Substituting \(g_c\) for g(t), we obtain
Substituting Eq. (12) for Eq. (8), we obtain
which is a dual exponential function. It is known that alpha and dual exponential functions fit well to biological data on synaptic potentials [16, 19, 27].
Figure 4 shows synaptic potential waveforms obtained from numerical simulation. Panels (a) and (b) show the waveforms with the alpha function (\(\tau _1=\tau _2\)), and (c) and (d) show those with the dual exponential function (\(\tau _1\ne \tau _2\)). To obtain panels (a) and (c) and panels (b) and (d), we changed \(\frac{t_w}{\tau _2}\) and the digital input code, respectively. In panels (a) and (c), the waveforms are so similar that they cannot be distinguished visually, and the peak of the potential does not change when \(\frac{t_w}{\tau _2}\) is varied. By contrast, as shown in panels (b) and (d), when the input code is \((10101010)_2\) or \((01010101)_2\), the peak changes. In the TDAC, \(C_{out}\) is not charged during \(S_{B,k}\) when memory bit k is zero, and this is caused by alternating charging and leaking. Moreover, the upper bit is converted into analog current faster than the lower one. However, we can ignore the influence of these when \(qt_w\) is sufficiently smaller than the time constant of the membrane potential of a neuron circuit.
4 Proposed Circuit
A block diagram of our proposed circuit is shown in Fig. 5, where an 8bit (sign+7) DAC can output negative and positive current. The circuit consists of a digital block and an analog block, the details of which are shown in Fig. 6.
The digital block consists of logic gates and delay circuits (DLs). Each DL consists of 11 transistors that output \(S_{B,k}\) with pulse width \(t_w\). The pulse width is the adjusted bias voltage \(V_{dl}\). If the bitlength of the TDAC increases, then an additional DL, two NAND gates, a NOT gate, two NMOS transistors, and two PMOS transistors are required, thereby increasing the total number of transistors by 25. The output signal of the digital block is the input for the analog block.
The analog block consists of voltagecontrolled current sources, switches, MOS resistors, and MOS capacitors. Negative and positive current outputs are realized by transistors \(\mathrm {M_{n}}\) and \(\mathrm {M_{p}}\), respectively. The number of transistors in the analog block does not increase when the bitlength of the TDAC increases. We explain the operation of positive and negative current output in Sections 4.2 and 4.3, respectively.
4.1 Positive Current Output
The proposed circuit outputs positive current when \(B_{8}\) is unity. As an example, Fig. 7a shows the timing diagram of the input signal and node voltages when the input digital bit code is \((11101000)_2\). The process for outputting positive current is as follows.

(1)
When \(S_{in}(t)\) turns high, \(S_{rsp}(t)\) and \(S_{stp}(t)\) turn high and low, respectively. At the same time, \(V_{gp}(t)\) is set to \(V_{rsp}\), where \(V_{rsp}\) is a small voltage.

(2)
When \(S_{in}(t)\) turns off, \(S_{B,7}(t)\) is generated at the trailing edge of \(S_{in}(t)\). At the same time, \(V_{gp}(t)\) increases exponentially.

(3)
\(S_{B,6}(t)\) is generated at the trailing edge of \(S_{B,7}\). If \(B_{6}\) is high, then capacitor \(C_{out}\) is charged during \(t_w\) by \(\mathrm {M_{p}}\).

(4)
Operation 3 is repeated until \(S_{B,1}(t)\) is generated.
We prevent wasteful power consumption by setting \(S_{stp}(t)\) to low while \(S_{in}(t)\) is high. This is because wasteful current flows from \(\mathrm {M_5}\) to \(\mathrm {M_4}\) if \(S_{stp}(t)\) is high while \(S_{in}(t)\) is high.
4.2 Negative Current Output
The proposed circuit outputs negative current when \(B_{8}\) is zero. Figure 7b shows the timing diagrams of the input signal and node voltages when the input digital bit code is \((01101000)_2\). The process for outputting positive current is as follows.

(1)
When \(S_{in}(t)\) turns high, \(S_{rsn}(t)\) and \(S_{stn}(t)\) turn low and high, respectively. At the same time, \(V_{gn}(t)\) is set to \(V_{rsn}\), where \(V_{rsn}\) is a high voltage.

(2)
When \(S_{in}(t)\) turns off, \(S_{B,7}(t)\) is generated at the trailing edge of \(S_{in}(t)\). At the same time, \(V_{gn}(t)\) decreases exponentially.

(3)
\(S_{B,6}(t)\) is generated at the trailing edge of \(S_{B,7}\). If \(B_{6}\) is high, then capacitor \(C_{out}\) is charged during \(t_w\) by \(\mathrm {M_{n}}\).

(4)
Operation 3 is repeated until \(S_{B,1}(t)\) is generated.
We prevent wasteful power consumption by setting \(S_{stn}(t)\) to high while \(S_{in}(t)\) is high. This is because wasteful current flows from \(\mathrm {M_9}\) to \(\mathrm {M_{10}}\) if \(S_{stn}(t)\) is low while \(S_{in}(t)\) is high.
5 Circuit Simulation
5.1 TimeDomain DigitaltoAnalog Conversion Without Leak Resistance
We designed an 8bit TDAC as shown in Fig. 5 with the TSMC 40 nm CMOS process (1 poly, 8 metal), and we evaluated the circuit by means of the Spectre simulation. We set the bias voltages and the capacitance as \(\mathrm {V_{DD}}=700\) mV, \(V_{dl}=180\) mV, \(V_{lkp}=140\) mV, \(V_{lkn}=420\) mV, and \(C_{out}=0.5\) pF, and the output voltage \(V_{out}(t)\) was reset to 350 mV on every input.
Figure 8 shows the input–output characteristics of the TDAC when \(V_{rsn}\) and \(V_{rsp}\) are varied separately. In the designed 8bit TDAC, the output current is negative when the digital input code is between zero and 128, and it is positive when the digital input code is between 129 and 255. The slopes of the characteristics in the two regions can be adjusted separately by varying \(V_{rsn}\) and \(V_{rsp}\) as shown in Fig. 8. The energy per one digitaltoanalog conversion is 27 fJ when the digital input code is \((11111111)_2\) and \(V_{rsp}=0\) mV.
5.2 TimeDomain DigitaltoAnalog Conversion With Leak Resistance
The circuit simulation for synapticpotential generation was conducted by adding a MOS resistance between the output node and the ground. We set the bias voltages and the capacitance as \(\mathrm {V_{DD}}=700\) mV, \(V_{dl}=340\) mV, \(V_{lkp}=300\) mV, \(V_{lkn}=300\) mV, \(V_{rstn}=570\) mV, \(V_{rstp}=70\) mV, and \(C_{out}=0.5\) pF.
Figure 9 shows the synapticpotential waveforms for various digital input codes \(S_{in}(t)\) and DL outputs. As shown in Fig. 4, which was obtained by numerical simulation, the waveforms are smooth when the lined bit is unity or zero, but in other cases the waveform has multiple peaks. We obtained waveforms that are similar to those from the numerical simulation.
5.3 Application of TimeDomain DigitaltoAnalog Converter to a Leaky IntegrateandFire Neuron Circuit
To demonstrate that the TDAC works as a synapse circuit of an SNN circuit, we applied the TDAC to a leaky integrateandfire (LIF) neuron circuit. Figure 10 shows an LIF neuron circuit and a block diagram of the LIF circuit with synapse circuits. The TDAC in this figure is the same circuit as shown in Fig. 5. In this situation, the digital input code to the mth TDAC represents a synaptic weight \(W_{m}\).
The neuron circuit in Fig. 10a integrates the currents from the TDACs by capacitor \(C_x\), and membrane potential \(V_x(t)\) is generated. The MOS transistor \(\mathrm {M_k}\) works as a leak resistor. Voltages \(V_{lk}\) and \(V_{rst}\) are the resting membrane potential and the reset potential, respectively (Fig. 10b). When \(V_x(t)\) reaches \(V_{th}\), output spike \(S_{out}(t)\) is generated and \(V_x(t)\) is reset to \(V_{rst}\). The bias voltages \(V_{bp1}\) and \(V_{bp2}\) adjust the firing threshold voltage \(V_{th}\) and the spike pulse width of \(S_{out}(t)\) by varying the rising time of \(V_{a}\) and \(V_{b}\). Voltage \(V_x\) is reset to \(V_{rst}\) when \(V_x(t)\) reaches \(V_{th}\). The current output nodes of the TDACs connect to \(V_x(t)\) as shown in Fig. 10c to achieve current–based summation asynchronously. Each TDAC has no resistor or capacitor for generating a PSP because the LIF circuit includes those components as \(\mathrm {M_k}\) and \(C_x\).
A circuit simulation with the LIF circuit was conducted. We used four TDACs as synapse circuits and set the bias voltages and the capacitance as \(\mathrm {V_{DD}}=700\) mV, \(V_{dl}=340\) mV, \(V_{lkp}=300\) mV, \(V_{lkn}=300\) mV, \(V_{rstn}=570\) mV, \(V_{rstp}=200\) mV, \(V_{rst}=200\) mV, \(V_{xlk}=200\) mV, \(V_{lk}=550\) mV, \(V_{bp1}=100\) mV, \(V_{bp2}=100\) mV, and \(C_{x}=0.5\) pF. The values of the synaptic weights are \(W_{1}=(11000000)_2\), \(W_{2}=(10100000)_2\), \(W_{3}=(01111000)_2\), and \(W_{4}=(11101011)_2\). Here, \(W_{3}\) is negative because the MSB is zero. The periods of input spikes \(S_{in,1}\), \(S_{in,2}\), \(S_{in,3}\), and \(S_{in,4}\) are 210 ns, 230 ns, 250 ns, and 270 ns, respectively.
The results of the circuit simulation are shown in Fig. 11. As shown in Fig. 11, a PSP is generated when the spike pulse arrives, and the neuron circuit generates spike pulse \(S_{out}(t)\) when \(V_x(t)\) reaches \(V_{th}\). The LIF circuit and the TDAC have no clock signal, and their operations start with the input spikes as their trigger; thus, the SNN circuit is an asynchronous system like biological neural networks. The circuit simulation results show that the TDAC can be used to implement neuromorphic hardware.
6 Conclusion
We proposed a new DAC, called a TDAC, in which the weight of each bit that codes for the DAC is realized by a current waveform sampled using nonoverlapping digital signals. The number of transistors needed to implement a TDAC increases linearly with the bitlength, but the transistors can be made small because they work as a digital circuit. A TDAC is therefore more compact than a conventional DAC in which the number of transistors increases exponentially with increasing bitlength. Moreover, a TDAC with leak resistance realizes biologically plausible synaptic responses without the need for other circuit components. TDACs are therefore suitable for implementing mixedsignal SNN hardware that requires high integration. Moreover, a circuit simulation showed that a TDAC works as a synapse circuit of SNNs by applying a TDAC to an LIF neuron circuit.
We showed the condition under which a TDAC remains linear, namely, that the ratio of the pulse width for sampling the current waveform to the time constant should be \(\ln 2\). To realize a TDAC that has good linearity and is robust against fabrication mismatches, the pulse width (resp. the time constant) should be set according to the time constant (resp. the pulse width), and we intend to develop such a circuit in our future work.
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Acknowledgements
This work was supported by the VLSI Design and Education Center (VDEC) of the University of Tokyo in collaboration with Cadence Design Systems, Inc. This work was also supported by the BMAI Project at IIS of the University of Tokyo.
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Uenohara, S., Aihara, K. TimeDomain DigitaltoAnalog Converter for Spiking Neural Network Hardware. Circuits Syst Signal Process 40, 2763–2781 (2021). https://doi.org/10.1007/s00034020015972
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DOI: https://doi.org/10.1007/s00034020015972