3. Hardware artificial neural networks

The hardware ANNs are based on a pulse-type hardware neuron model of class II, specifically, a Hodgkin-Huxley model and a Bonhoeffer-van der Pol model [28]. This section describes the circuit diagrams and the basic characteristics of the pulse-type hardware neuron model. The synchronization phenomena of the hardware ANNs are also discussed.

#### 3.1. Pulse-type hardware neuron model

Figure 6 shows the circuit diagrams of the pulse-type hardware neuron model, which comprises a cell body model and two synaptic models. The cell body model (Figure 6a) includes a voltage control-type negative resistance, an equivalent inductance, resistors R<sup>1</sup> and R2, and a membrane capacitor CM. The voltage control-type negative resistance circuit with equivalent inductance consists of an n-channel MOSFET M1, a p-channel MOSFET M2, a voltage source VA, a leak resistor RL, another resistor RG, and a capacitor CG. The cell body model generates oscillating patterns of electrical activity vM (t). vG (t) is the voltage between both ends of

Figure 6. Circuit diagram of the pulse-type hardware neuron model. (a) Cell body model, (b) excitatory synaptic model, and (c) inhibitory synaptic model.

capacitor CG. vM (t) and vG (t) are, respectively, governed by the following simultaneous differential equations.

Figure 5 shows the relative phase differences of the hexapod gait patterns for different driving rhythms of the actuator. To heat the artificial muscle wires, an input pulse of amplitude 50– 100 mA, period 2 s, and width 0.5 s is required. Therefore, the hexapod robot requires 2 s to complete one locomotion cycle. As mentioned above, the length of the artificial muscle wire depends on the temperature. Specifically, the wire shrinks when heated and extends when cooled. Heating the artificial muscle wires from A to D and from D to A in Figure 4 drives the hexapod robot forward and backward, respectively. The locomotion pattern is a 180 phase shift at each side, which mimics the locomotion of an ant. If the input pulse is narrower than 0.5 s, the thermal heating by the driving current is insufficient to shrink the wire. By contrast, if the input pulse is wider than 2 s, the thermal heating by the driving current is excessive, and

The hardware ANNs are based on a pulse-type hardware neuron model of class II, specifically, a Hodgkin-Huxley model and a Bonhoeffer-van der Pol model [28]. This section describes the circuit diagrams and the basic characteristics of the pulse-type hardware neuron model. The

Figure 6 shows the circuit diagrams of the pulse-type hardware neuron model, which comprises a cell body model and two synaptic models. The cell body model (Figure 6a) includes a voltage control-type negative resistance, an equivalent inductance, resistors R<sup>1</sup> and R2, and a membrane capacitor CM. The voltage control-type negative resistance circuit with equivalent inductance consists of an n-channel MOSFET M1, a p-channel MOSFET M2, a voltage source VA, a leak resistor RL, another resistor RG, and a capacitor CG. The cell body model generates oscillating patterns of electrical activity vM (t). vG (t) is the voltage between both ends of

Figure 6. Circuit diagram of the pulse-type hardware neuron model. (a) Cell body model, (b) excitatory synaptic model,

synchronization phenomena of the hardware ANNs are also discussed.

the cooling is insufficient to extend the wire.

34 Advanced Applications for Artificial Neural Networks

3. Hardware artificial neural networks

3.1. Pulse-type hardware neuron model

and (c) inhibitory synaptic model.

$$\mathbf{C}\_{M}\frac{d\upsilon\_{M}(t)}{dt} = \frac{\upsilon\_{s}(t)}{R\_{in}} - \frac{\upsilon\_{M}(t)}{R\_{L}} - \frac{\upsilon\_{M}(t) - \upsilon\_{G}(t) - V\_{A}}{R\_{G}} + i\_{A}(\upsilon\_{M}(t), \upsilon\_{G}(t)),\tag{1}$$

$$\mathbb{C}\_{\mathcal{G}} \frac{d\upsilon\_{\mathcal{G}}(t)}{dt} = \frac{\upsilon\_{\mathcal{M}}(t) - \upsilon\_{\mathcal{G}}(t) - V\_{A}}{R\_{\mathcal{G}}}.\tag{2}$$

A time-dependent nonlinear current iΛ(vM(t),vG(t)) flows through the negative resistance circuit. The governing equations of iΛ(vM(t),vG(t)) under the three possible conditions are given by the following:

$$\text{Condition 1}: \upsilon\_{\mathbb{G}}(t) + V\_A + V\_{\mathbb{T}n} + V\_{\mathbb{T}p} < \upsilon\_{\mathbb{M}}(t) \le V\_{\mathbb{G}p} + V\_{\mathbb{T}p}.$$

$$i\_{\Lambda}(\upsilon\_{\mathbb{M}}(t), \upsilon\_{\mathbb{G}}(t)) = \frac{\beta}{8} \left( A + B + V\_{\mathbb{G}p} \right)^2\tag{3}$$

Condition 2 : VGp þ VTp < vMð Þt ≤ vGð Þþ t VA þ VTn,

$$\dot{v}\_{\Lambda}(\upsilon\_{M}(t), \upsilon\_{G}(t)) = \frac{\beta \cdot A^{2} \left(A - 2\left(B + V\_{Gp}\right)\right)^{2}}{8\left(A + B + V\_{Gp}\right)^{2}}\tag{4}$$

$$\text{Condition 3}: \upsilon\_G(t) + V\_A + V\_{Tn} < \upsilon\_M(t) \le V\_{A\nu}$$

$$i\_{\Lambda}(\upsilon\_{M}(t), \upsilon\_{G}(t)) = $$

$$\frac{\frac{\beta \left\{ (V\_{A} - \upsilon\_{M}(t))(V\_{A} - \upsilon\_{M}(t) + 2A) \left(V\_{A} + \upsilon\_{M}(t) - 2\left(V\_{Tp} + V\_{Gp}\right)\right) \left(V\_{A} - 3\upsilon\_{M}(t) + 2\left(V\_{Tp} + A + V\_{Gp}\right)\right) \right\}}{8\left(A + B + V\_{Gp}\right)^{2}} \tag{5}$$

where <sup>A</sup>¼vGð Þþ<sup>t</sup> VTn,B¼VTp�vMð Þ<sup>t</sup> ,VGp<sup>¼</sup> VA∙R<sup>2</sup> R1þR<sup>2</sup> , β 2¼με 2t W <sup>L</sup> ,and VTn and VTp are the threshold voltages of the n and p-channel MOSFETs, respectively. VGp is the gate voltage of MOSFET M2. β is the conductance constant of the MOSFETs (with carrier mobility μ, dielectric constant ε of the gate insulator, oxide channel thickness t, channel width Wand channel length L). Although the value of β differs in the n-type and p-type MOSFETs, Eqs. (3–5) become intractable unless the βs are approximated by the same value. The complex case with different βs is considered in the following numerical analysis.

The circuit parameters of the cell body model are as follows: CG = 4.7 μF, CM = 470 nF, R<sup>G</sup> = 680 kΩ, R<sup>L</sup> = 10 kΩ, R<sup>1</sup> = 15 kΩ, R<sup>2</sup> = 20 kΩ, and Rin = 50 kΩ. The voltage source VA = 3.5 V. The authors have used the BSS83 and BSH205 for M<sup>1</sup> and M2, respectively. These circuit parameters are set to allow the cell body model to generate oscillation with amplitude 3.5 V, period 8 s, and width of 2 s.

Figure 6b and c displays the circuits of the excitatory and inhibitory synaptic models, respectively. The spatiotemporal summation characteristics of the synaptic model resemble those of biological systems. The output vS(t) of the synaptic model is the spatiotemporal summation of the output voltages of the cell body model vM(t). vES (t) and vIS (t) are described by the following equations.

$$
\upsilon\_{ES}\left(t\right) = \frac{R\_{ES2}}{R\_{ES1}} \cdot \frac{R\_{ES4}}{R\_{ES3}} \left(1 - e^{-\frac{t}{R\_{ES2} \cdot C\_{ES}}}\right) \cdot \upsilon\_M(t),\tag{6}
$$

$$\upsilon\_{l\mathcal{S}}\left(t\right) = -\frac{R\_{l\mathcal{S}2}}{R\_{l\mathcal{S}1}} \left(1 - e^{-\frac{t}{R\_{l\mathcal{S}2} \cdot \mathcal{C}\_{l\mathcal{S}}}}\right) \upsilon\_{\mathcal{M}}(t). \tag{7}$$

The spatial summation is performed by the inverting amplifier, whose amplification factor imitates the synaptic weight. Suffixes E and I denote excitatory and inhibitory, respectively. The temporal summation is realized by the operational amplifier RC integrator. The resistors and capacitors in the synaptic model are valued at 1 MΩ and 1 pF, respectively. The operational amplifier is an RC4558D.

#### 3.2. Basic characteristics of the cell body model

Figure 7 shows the negative resistance characteristics of the cell body model. The N-shape characteristic indicates that the negative resistance is voltage control type negative resistance. When 2.3 < vM (t) < 3.5 (vG(t) = 2.5 V), the negative resistance is provided by the negative resistance circuit. The amplitude of the negative resistance characteristic can be changed by varying the vG(t).

Figure 8 shows the phase plane of the cell body model. The attractor (solid line in Figure 8) is the limit cycle. The shapes of the vM(t)- and vG(t)-nullclines confirm the class II neuron characteristics of the cell body model. The same characteristics are observed in the Hodgkin-Huxley and Bonhoeffer-van der Pol models. The vM(t)- and vG(t)-nullclines intersect at the equilibrium point. When the equilibrium point dvM/dvG > 0, the cell body model becomes unstable and selfoscillates. By contrast, when dvM/dvG < 0, the cell body model becomes stable. In this chapter, the equilibrium point is set to the unstable condition dvM/dvG > 0. The unstable and stable conditions can be switched by varying VA.

#### 3.3. Excitatory-inhibitory neuron pair model

The excitatory-inhibitory neuron pair model comprises two cell body models and two synaptic models and generates several oscillatory patterns by using the synchronization phenomena.

Figure 7. Negative resistance characteristic of the cell body model. The abscissa is vM (t) and the ordinate is iΛ(vM(t), vG(t)).

Figure 9 shows the circuit diagram of the excitatory-inhibitory neuron pair model. The cell body models are mutually coupled by the synaptic models. The excitatory synaptic model sums the excitatory inputs (output voltage of the cell body model vME and the external input voltage vextinE). Meanwhile, the inhibitory synaptic model sums the inhibitory inputs (output voltage of the cell body model vMI and the external input voltage vextinI). Both cell body models are assigned the same circuit parameters (The synchronization phenomena and oscillatory patterns of the mutually coupled excitatory-inhibitory neuron pair model are provided in [28]).

vES ðÞ¼ t

tional amplifier is an RC4558D.

36 Advanced Applications for Artificial Neural Networks

3.2. Basic characteristics of the cell body model

conditions can be switched by varying VA.

3.3. Excitatory-inhibitory neuron pair model

vIS ðÞ¼� t

RES<sup>2</sup> RES<sup>1</sup> ∙ RES<sup>4</sup> RES<sup>3</sup>

> RIS<sup>2</sup> RIS<sup>1</sup>

1 � e

1 � e

The spatial summation is performed by the inverting amplifier, whose amplification factor imitates the synaptic weight. Suffixes E and I denote excitatory and inhibitory, respectively. The temporal summation is realized by the operational amplifier RC integrator. The resistors and capacitors in the synaptic model are valued at 1 MΩ and 1 pF, respectively. The opera-

Figure 7 shows the negative resistance characteristics of the cell body model. The N-shape characteristic indicates that the negative resistance is voltage control type negative resistance. When 2.3 < vM (t) < 3.5 (vG(t) = 2.5 V), the negative resistance is provided by the negative resistance circuit. The amplitude of the negative resistance characteristic can be changed by varying the vG(t). Figure 8 shows the phase plane of the cell body model. The attractor (solid line in Figure 8) is the limit cycle. The shapes of the vM(t)- and vG(t)-nullclines confirm the class II neuron characteristics of the cell body model. The same characteristics are observed in the Hodgkin-Huxley and Bonhoeffer-van der Pol models. The vM(t)- and vG(t)-nullclines intersect at the equilibrium point. When the equilibrium point dvM/dvG > 0, the cell body model becomes unstable and selfoscillates. By contrast, when dvM/dvG < 0, the cell body model becomes stable. In this chapter, the equilibrium point is set to the unstable condition dvM/dvG > 0. The unstable and stable

The excitatory-inhibitory neuron pair model comprises two cell body models and two synaptic models and generates several oscillatory patterns by using the synchronization phenomena.

Figure 7. Negative resistance characteristic of the cell body model. The abscissa is vM (t) and the ordinate is iΛ(vM(t), vG(t)).

� <sup>t</sup> RES2∙CES 

� <sup>t</sup> RIS2∙CIS  ∙vMð Þt , (6)

∙vMð Þt : (7)

Figure 8. Phase plane of the cell body model. The abscissa is vM(t) and the ordinate is vG(t). The dotted, broken, and solid lines display the vM(t)-nullcline, vG(t)-nullcline, and attractor, respectively.

Figure 9. Circuit diagram of the excitatory-inhibitory neuron pair model.

Figure 10. Four sets of excitatory-inhibitory neuron pair models connected by an inhibitory synaptic model. (a) Connection diagram, (b) output waveform when the external trigger pulse generates a walk sequence, (c) output waveform when the external trigger pulse generates a trot sequence (The waveforms in panels (b) and (c) are the simulation results).

The synchronization phenomena of the excitatory-inhibitory neuron pair circuit depend on the connection type of the synaptic model. When the synaptic model connection is excitatory or inhibitory, the synchronization is in-phase and anti-phase, respectively. The excitatory-inhibitory neuron pair circuit was incorporated into the hardware ANN for the quadruped robot.
