We conducted our experiments using a combination of computational simulations and hardware implementations. The computational simulations were carried out using software tools such as Protes (ISIS), where we developed mathematical models representing the electrical circuits analogous to neural networks. These models incorporated parameters reflecting the dynamics of neurons, synapses, and their interactions.
For hardware implementations, we utilized electronic components to construct physical circuits mimicking the behavior of neuronal networks. These circuits consisted of operational amplifiers, resistors, capacitors, and other discrete elements interconnected to emulate the synaptic connections and neuronal firing patterns observed in the brain.
We use a Chua circuit to investigate chaotic dynamics in neural networks, which is the “Chua circuit”.
We will add other things to this circuit as shows on Fig. 1.
In this circuit, elements R1 and R2 represent resistors, C1 and C2 are capacitors, L stands for an inductance, and G(x) is a nonlinear function. Voltages V1 and V2 serve as the circuit's inputs, while Vout represents the output voltage. We have chosen G(x) to be the FitzhughNagumo function.
a. Chua Circuit:
Chua's Circuit is a simple electronic circuit that exhibits chaotic behavior, making it suitable for studying complex dynamics like those observed in neural networks (Fig. 2).
The dynamics of the circuit are defined by the following equations:
$$\left\{\begin{array}{c}{\varvec{C}}_{1}{\dot{\varvec{V}}}_{1}=\frac{{\varvec{V}}_{2}{\varvec{V}}_{1}}{\varvec{R}}f\left({\varvec{V}}_{1}\right)\\ {\varvec{C}}_{2}{\dot{\varvec{V}}}_{2}=\frac{{\varvec{V}}_{1}{\varvec{V}}_{2}}{\varvec{R}}+i \\ L\dot{\dot{\varvec{i}}}={\varvec{V}}_{2 } \end{array}\right.$$
(1)
V₁ and V₂, representing the voltage, capacitors C₁ and C₂, i denoting the current flowing through the inductance, and f representing the current response of the Chua diode.
Figure 2 shows the diagram Chua circuit.
b. Diode Tunnel:
The use of tunnel diodes in neural networks is primarily due to their unique electrical properties, which can be exploited to implement specific functionalities within the network architecture. One of the key advantages of tunnel diodes is their ability to exhibit negative differential resistance (NDR) behavior. (Fig. 3)
This property can be leveraged to introduce nonlinearity into the network's activation functions. In neural networks, nonlinearity is crucial for enabling complex computations and learning processes.
Also, tunnel diodes offer fast switching speeds and low power consumption, which are desirable traits for implementing efficient and highperformance neural network hardware. Their suitability for highfrequency operation also makes them promising candidates for neuromorphic computing applications, where hardware mimics the brain's biological neural networks.
c. FitzhughNagumo function:
The FitzHughNagumo function is a mathematical model used to describe the dynamics of excitable systems, particularly in the context of neuroscience. It consists of a system of ordinary differential equations that capture the behavior of a neuron's membrane potential and recovery variable.
The FitzhughNagumo model is often represented by the following equations:
$$\frac{dV}{dt}=V\frac{{V}^{3}}{3}W+I$$
$$\frac{dW}{dt}=Є(V+abW)$$
(2)
where V represents the membrane potential of the neuron, W is the recovery variable, I is the input current, a is a parameter representing the threshold for spike initiation, b represents the sensitivity of the recovery variable to changes in the membrane potential, and epsilon(ϵ) is a small parameter determining the time scale separation between V and W.
Here's how you could implement this model in an electronic circuit:

Circuit for V: we can use an operational amplifier (opamp) in comparator mode with positive feedback to simulate the behavior of the variable V. Nonlinear elements, such as diodes, can be used to introduce the necessary nonlinearities to simulate the term V3/3.

Circuit for W: we use another operational amplifier with an appropriate configuration to simulate the behavior of the variable W. You can adjust the parameters of this circuit to match the values of a, b, and ε.

Input Current Source (I): we can add a current source to simulate the input current I in the model. This current source can be a voltagecontrolled current source or a currentcontrolled voltage source, depending on your requirements.

Interaction between V and W: The components of the circuit for V and W should be interconnected appropriately to reflect the interactions described in the differential equations.

Make sure to provide appropriate power to the circuit and configure initial conditions to simulate the initial behavior of the system.
Figure4 shows the circuit diagram of the FitzHugh − Nagumo Function.