Journal of Robotics, Networking and Artificial Life, Vol. 5, No. 1 (June 2018) 71–74___________________________________________________________________________________________________________Finding appropriate parameter voltages for driving a low-power analog silicon neuron circuit
Atsuya Tange

Department of Electrical Engineering and Information Systems, The University of Tokyo, 4-6-1 Komaba,
Meguro-ku, Tokyo, 153-8505, Japan
Takashi Kohno

Institute of Industrial Science, The University of Tokyo, 4-6-1 Komaba,
Meguro-ku, Tokyo, 153-8505, Japan
E-mail: tange@sat.t.u-tokyo.ac.jp, kohno@sat.t.u-tokyo.ac.jp
www.u-tokyo.ac.jp

Abstract
This research focuses on a silicon neuron circuit designed utilizing a qualitative neuronal modeling approach. In thiscircuit, temperature, fabrication mismatch, and secondary effects of transistors cause the difference between theintended characteristics and those in the implemented circuits. Therefore, we have to tune the bias voltages for eachneuron instance to realize the desired dynamical behavior after circuit implementation. We constructed an algorithm
to automatically find appropriate values for the bias voltages.

Keywords: neuromorphic hardware, neuromorphic chip, silicon neurons, analog VLSI

utilizing the qualitative modeling techniques, the model ofthis circuit was designed so that it can reproduce thedynamical structures in the excitable nerve membrane.
The parameter tuning procedure is supported by thefeedback amplifiers integrated in the silicon neuron circuit.
The characteristics of this circuit are influenced bytemperature, transistors’ fabrication mismatch, and theirsecondary effects (short channel effect, etc). They makethe expected circuit characteristics in designing stage andactual behavior of the individual circuit inconsistent.
Therefore, it is required to adjust bias voltages oftransistors in silicon neuron circuit to get the desired
dynamical behavior.
The procedure to do it by hand has been established3,
but it is not realistic to execute the procedure for eachsilicon neuron circuit when we build the large-scale siliconneuronal networks comparable to the human brain whichcontains about 100 billion neurons.

1. Introduction
Silicon neuron is electronic circuit that mimics
electrophysiological behavior of neurons. It is supposed to
be used as the basic elements of silicon neural networks,
whose aim is to simulate the behavior of the nervous
system in real-time or faster. They can be used not only as
the high-speed brain simulators for neuroscientificresearches but also as a basic technology for the next
generation low-power intelligent computing systems. In
addition, they can be used to construct bio-silico hybrid
systems in connection with neurons, which can be an ideal
technology for neuroprosthetic devices.
In this research, we focus on an analog silicon neuron
circuit designed by using the techniques of qualitative
neuronal modeling whose power consumption is as low as
about 3 nW.1 This circuit is composed of metal-oxidesemiconductor field-effect transistors (MOSFETs) in theirsubthreshold region for low-power consumption. ByCopyright © 2018, the Authors. Published by Atlantis Press.
This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
71

Journal of Robotics, Networking and Artificial Life, Vol. 5, No. 1 (June 2018) 71–74___________________________________________________________________________________________________________Fig. 2. Automatic bias voltages tuning system forsilicon neuron

has shallower gradient than the former. They are combined
to construct the mathematical structures in the neurons.
andare constant current and input stimulus,respectively. They are generated by transconductance
amplifiers.
There are nine bias voltages to tune this circuit, one ofwhich is related to the time constant of and the others
affect the shape of the sigmoidal curves.
Our circuit can reproduce Class1 and 2 neurons in the
Hodgkin’s classification6, by reconstructing specificdynamical structures for each class. In this work, weconcentrated on the Class1 neuron mode of our circuit.
To estimate the dynamical structures in our circuit, thetransconducance amplifiers (TAV and TAN in Figure 1)
are used to draw the nullcline of each variable. Thenullclines are drawn by measuring the current, theoutput current of TAV, and , the output current of,while sweepingin DC analysis. Their equations arerepresented as below.
(3)Fig. 1. Block diagram of our qualitative-modelingbased low power silicon neuron circuit

To solve this issue, a system to fit characteristics ofsilicon neuron automatically is essential. In this work, thesame metaheuristic algorithm used in the previous similarworks2, 3, 4, the Differential Evolution (DE) method5, isused to find proper bias voltages. The DE method
performed better in terms of convergence speed andsimulation time, compared to other popular metaheuristic
algorithms, such as Genetic Algorithm and Simulated
Annealing.3 All the results in this work was obtained bycircuit simulation using ngspice.
Our silicon neuron model is explained in the next
section and our parameter fitting algorithm is in section 3.
The results and discussion are in section 4 and 5,respectively.

(4)

2. A low-power analog silicon neuron model
3. Automatic parameter voltages fitting
algorithm

Our silicon neuron circuit is composed of two blocks, the- and -blocks (Figure 1). Each block’s dynamics isrepresented as below.
,,

Figure 2 illustrates the block diagram of our automaticparameter fitting system. Here, we use circuit simulation
instead of actual circuit measurement because this work
focuses on the efficient verification of our fitting
algorithm in this system. It finds the appropriate parametervalues by optimizing the difference between the reference
data and the simulation result.
This optimization is done by using the DE method
which searches for the best-fitted parameter vector byrepeating the mutation, crossover, and selection ofparameter vectors for a number of generations. The(1)(2)

where and represents the membrane potential and thedynamics of ionic channels, respectively.
For each function,,is an ideal I-V
characteristics of a differential pair circuit,andare ideal I-V characteristics of a cascoded transistor
circuit with source degeneration with a detached bulk
voltage. Both of them are in sigmoidal shape and the latter72

Journal of Robotics, Networking and Artificial Life, Vol. 5, No. 1 (June 2018) 71–74___________________________________________________________________________________________________________to produce the desired behavior similar to the reference
model, because our circuit’s dynamical properties are
sensitive to the shape of the nullclines and rn_Vm cannot
be fitted in Step1. In this step, we fit the dynamical
behavior in response to stimulus using the DE method.
Threshold current, spike width in response to pulse
stimulus, and spike frequency in response to sustained
stimulus are used for fitness function.
The definition of the threshold current is the minimum
amplitude of the pulse stimulus that makes the membrane
voltage exceed 400mV. The spike width is defined as time
width when the membrane voltage at the middle of the
spike amplitude.
Our silicon neuron circuit is equipped withtransconductance amplifiers for generating current
stimulus. Pulse stimulus with 500
time width wasgenerated by applying a pulse voltage to thetransconductance amplifier. We applied pulse stimuluswith a variety of amplitude by increasing the pulse voltage
by 7mV step for 20 pulses. The threshold current and the
spike width were measured with the firstly observed spikein this sequence.
For the spike frequency, two amplitudes of sustained
stimulus, 10pA and 15pA, were applied for 300ms using
the transconductance amplifier.
All of the 9 parameters are tuned in this step. The initial
parameter value was distributed in the range within thevalue obtained by Step1 2 mV except for rn_Vm. For
rn_Vm, an initial value was found by sweeping until
spikes are generated in response to both pulse andsustained stimulus with the other parameters’ value
obtained by Step1. The range for this sweeping was
between the reference model’s parameter value (435 mV)-6mV to +12mV.
In this step, the fitness function is calculated as follows:Fig. 3. Nullclines in the Class1 neuron mode

automatic parameter fitting algorithm we suggest iscomposed of the 2 steps explained below.
3.1. Step1: Fitting nullclines
At first, we fit the shape of the - and -nullclines to agiven reference nullclines. Figure 3 shows the nullclines
in the Class1 neuron mode without stimulus. The twonullclines intersect each other at three points, “S”, “T”,
and “U”. They represent a stable node, a saddle point, andan unstable node, respectively. Point “S” corresponds to
the resting state and “T” is responsible for the thresholdphenomena of neuronal spike generation.
The fitness function for the DE method is calculated bysumming the difference between the nullcines obtained in
the simulation and the reference nullclines. The fitness
function is written as follows:∑,,(5)
1, 2 ⋯ ,75,, ,and , represent the coordinate of a point on
wherethe nullclines in simulation result and the referencenullclines, respectively. is index of each point andrepresents or . The voltage range of the nullclines islimited to between 300 and 450 mV. This range covers thethree intersections of the nullclines in the Class1 neuronmode. Four parameters, fx_Vb, fx_Vdlt, gx_Vm, Iax_Vin(x= , ) are tuned for each nullcline. Parameter rn_Vm isfixed at a random value in this step.
Initial vectors are generated in the range between 100mV and +100mV compared to the values used forgenerating the reference data. The DE method’s
optimization step was repeated for 400 generations and 40
vectors were contained in each generation.

||,

where,1,3.2. Step2: Fitting neuron characteristics

,(6)

,,.

is obtained by each transient simulation result andis obtained by the reference model. Elements of weight

The nullcline fitting procedure in Step1 was not sufficient73

Journal of Robotics, Networking and Artificial Life, Vol. 5, No. 1 (June 2018) 71–74___________________________________________________________________________________________________________Table 1. Comparison of silicon neuron characteristics
Error (%)
Reference
Tuned-0.953
262.1
259.6
pA-3.30
18.0
17.4
[ms]
29.930.8+3.01
[Hz]
[Hz]

42.9

43.5

+1.40

vector, , are calculated by dividing
by elementsof , respectively.
In this step, the DE method’s step was repeated for
200 generations and 45 vectors were contained in eachgeneration.

Fig. 4. Transient data of the reference (top) and thesimulation results of the tuned circuit (bottom)

neuronal classes such as the regular spiking, the squarewave bursting, and the elliptic bursting.
In the future, we will apply this algorithm to a realcircuit.

4. Results
Table1 shows the characteristic of our silicon neuron
circuit with the parameters obtained by our fitting
algorithm. For all the criteria, the error is less than 4%.
Figure 4 shows the reference transient data andsimulation results of the fitted circuit. The amplitude of
spikes seems to be well fitted.
The reference data was generated by using ngspicesimulation. Thus the ideal fitting result is identical to theparameter set used for the reference data generation, withwhich the error is zero. But as shown in Table 1, ouralgorithm could not find the original parameter set but
found a local optimum point.

Acknowledgement
This study was partially supported by JST PRESTO andCREST and JSPS-MAEDI SAKURA Program.
References
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5. Discussion
In this research, we proposed an algorithm to find properparameter values to obtain the desired dynamical
properties for individual silicon neuron circuit instance.
This algorithm will be crucial for operation of large-scale
silicon neuron networks. The error for the criteria in thefitting procedure was less than 4%. From the viewpoint ofengineering application, defining the maximum
acceptable error is a difficult problem, because it depends
on specific applications but they are not established yet.
From the viewpoint of neuromimetics, this error isacceptable when our algorithm is used to tune a number ofsilicon neuron instances to Class 1, because there is a wide
distribution (far larger than 4%) of dynamical properties
in the same class of neuronal cells.
We will also try to apply this algorithm in fitting Class2
neuron. In addition, our algorithm will be improved to be
applicable to the three variable ultralow-power siliconneuron circuit.7 This circuit can realize a wide variety of74