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It is shown how to implement by a mixed-signal circuit a continuous -time dynamical system. The chosen case study is the Hindmarsh-Rose model of a biological neuron, but the design strategy can be applied to a large class of continuous-time nonlinear dynamical systems. The system nonlinearities are first approximated by using piecewise-linear functions and then digitally implemented on a field programmable gate array. The linear part of the system is completely analogue and is implemented by using operational amplifiers. Measurement results show that the circuit can reproduce the main dynamics of a biologically plausible neuron.
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Introduction: In [1] it was shown that, in principle, any autonomous nonlinear dynamical system described by a set of ordinary differential equations (ODEs) of the kind λ: = f(x\ p) can be approximated by replacing the (smooth) vector field with a proper piecewise-linear (PWL) vector field. Here x G R" (state vector), /> G R? (parameter vector), / : S C R"+î -* R" (vector field), S is a limited compact domain, and x denotes the derivative of x with respect to time. Since there are many digital architectures that implement (by linear interpolation) multivaried nonlinear functions [2], this allows realisation of quite a large class of dynamical systems by circuits.
In this Letter, we provide a case study that can be viewed as a proof of concept of the PWL approximation/implementation paradigm. We have chosen to implement the biologically plausible neuron model proposed by Hindmarsh and Rose [3], The two main advantages in using this model are: (i) it is relatively simple, and (ii) there exist circuit syntheses [4, 5], Therefore this model is a good benchmark to test the proposed method, in view of both a circuit implementation of the HindmarshRose (HR) model and the application of the whole approximation/synthesis procedure to more complex (and physiologically realistic) neuron models, such as the Hodgkin-Huxley one [6], Moreover, the circuit implementation of neuron models could open new perspectives in the field of simulation of biologically plausible neural networks, one of the most ambitious challenges taken up by the international scientific community. The first step towards this direction is the circuit synthesis of the elementary unit of the network.
The circuit is implemented on...