Circuit Realization of the Inertia Neuron

Zden k HRUBOŠ, Tomáš GÖTTHANS, Ji í PETRŽELA

Dept. of Radio Electronics, Brno University of Technology, Purky ova 118, 612 00 Brno, Czech Republic

xhrubo00@stud.feec.vutbr.cz, xgotth00@stud.feec.vutbr.cz, petrzelj@feec.vutbr.cz

Abstract. In this article a simple model of inertia neuron based on the ordinary differential equations of Hindmarsh-Rose model is shown. The fully analog circuit realization is presented. Operational amplifiers as an inverting integra-tor are used for integration of the differential equations. The function of the model is checked by varying of five bifurcation parameters. Simulations and measurements results are compared. The simulation and measured results demonstrate that analog neuron circuit model is the viable candidate for the implementation in real time.

Keywords Neuron models, Hindmarsh-Rose model, differential

equations, membrane potential, time domain, plane projection.

1. Neuron Models Neural models are used in both computational neuro-

science and in pattern recognition. In both cases, the highly parallel nature of the neural system contrasts with the sequential nature of computer systems, resulting in slow and complex simulation software. More direct hardware implementation holds out the promise of faster emulation both because it is inherently faster than software and the operation is much more parallel. In fact, direct hardware implementation of neural models has a relatively long history.

Nowadays a lot of mathematical neuron models exist. One of the first used for mathematical describing of the neuron behavior was Hodkin-Huxley model. This model explains the ionic mechanisms underlying the initiation and propagation of action potentials in the squid giant axon. The simplified version of the Hodgkin–Huxley is a Fitz-Hugh–Nagumo (FHN) model. FHN model is describing in a detailed manner activation and deactivation dynamics of a spiking neuron. The other neuron model conception building upon the FitzHugh-Nagumo model is from J. L. Hindmarsh and R. M. Rose. They proposed a Hindmarsh-Rose (HR) model of neuronal activity is described by three coupled first order differential equa-tions. This extra mathematical complexity allows a great variety of dynamic behaviors for the membrane potential, described by the x variable of the model, which include

chaotic dynamics. This makes the Hindmarsh-Rose neuron model very useful, because being still simple, allows a good qualitative description of the many different patterns of the action potential observed in experiments [1], [2].

1.1 FitzHugh–Nagumo Model One of the simplest models used for mathematical

describing of a prototype an excitable system (e.g., a neuron) is FitzHugh-Nagumo model. Fitz Hugh and Nagumo seek to do to describe "regenerative self-excitation" by a nonlinear positive-feedback membrane voltage and recovery by a linear negative-feedback gate voltage, they developed the model described by

byax

dtdy

Iyxxdtdx

ext ,3

(1)

where we again have a membrane-like voltage and input current with a slower general gate voltage w and experimentally-determined parameters.

If the external stimulus Iext exceeds a certain threshold value, the system will exhibit a characteristic excursion in phase space, before the variables v and w relax back to their rest values. This behavior is typical for spike generations (=short elevation of membrane voltage v) in a neuron after stimulation by an external input current [1].

1.2 Hindmarsh-Rose Model The neuronal activity of Hindmarsh-Rose (HR) model

is study the spiking-bursting behavior of the membrane potential. Observation is aimed to experiments made with a single neuron.

The relevant variable is the membrane potential, x(t). Other variables are y(t) and z(t), which take into account the transport of ions across the membrane through the ion channels. Variable y(t) is called the spiking and is describ-ing transport of sodium and potassium ions through fast ion channels. Variable z(t) is called the bursting and its func-tion is in transport of other ions thorough slow channels [3].

Hindmarsh-Rose model is determined by a system of three nonlinear ordinary differential equations with

978-1-61284-324-7/11/$26.00 ©2011 IEEE

dynamical variables x(t), y(t), and z(t). ODEs have the following form

Izxydtdx ,

yxdtdy , (2)

zxxbdtdz

0

where

32 xaxx , (3)

21 dxx . (4)

How we can see the model has six parameters: a, b, d, , xr and I. The roles of the individual system parameters

are the following:

I represent the membrane input current for biological neurons.

b controls switching between bursting and spiking behaviors and allows to control the spiking frequency.

is rate of change of the slow variable z. For spiking behaviors, allows controls the spiking frequency. For bursting behaviors the number of spikes per burst is influenced by .

d governs adaptation. A unitary value of d determines spiking behavior without accommodation and sub threshold adaptation. Whereas, around d = 4 give strong accommodation and sub threshold overshoot, or even oscillations.

x0 sets the initial conditions of the system [3].

The practical experiments show that it is very com-mon to fix some of them and let the other to be control parameters. Parameter I is the most common parameter used for controlling of HR model function and is simulated the current that enters the neuron. Other control parameters used in HR model have the following functions: parameters a and b are modeling the working of the fast ion channels and the parameter r is modeling the working of the slow ion channels. Typical values of fixed parameters are: a = 3, b = 5, d = 4, xr = 8/5. The parameter is something of the order of 10 3, and range of I is between 10 and 10.

2. Circuit Implementation of HR Model For practical implementation the most straightforward

method known as integrator based synthesis is used [4], [5], [7], [8]. Method is based on knowledge of the mathe-matical model [3], [6] of a dynamical system. The advan

tage of this method is in the low number of necessary functional blocks - inverting integrator, differential (or summing) amplifier and two two-port blocks with desired nonlinear transfer function. State variables are represented by the output voltage of integrators and therefore are easily measurable. Parasitic properties of the active components are not critical because the time constant circuit can be selected in the audio band. The drawback is also evident: a large number of passive and active circuit elements.

System based on the ordinary differential equations of HR model (2) is presented. The default block diagram of the inertia neuron with three integrators and two multipliers is shown in Fig. 1. Circuitry implementation of mathemati-cal HR model is using three operational amplifiers TL084 for realizations of inverting integrator and one operational amplifier for realizations summing amplifier. The nonlinear two-port circuit is formed by a connection of two four-quadrant analog multipliers AD633 with transfer function

ZYYXXW UUUUUKU 2121 (5)

where K = 0.1 is constant [9].

Fig. 1. Block diagram of the inertia neuron with three

integrators and two multipliers.

3. Experimental Verification The functionality of circuit implementation of inertia

neuron was first successfully tested in PSpice simulator. Simulated plane projections associated with a designed of inertia neuron are shown in Fig. 2. Correct function of the dynamical system was verified also experimentally. Plane projections and frequency spectrum of the selected signals measured by means of Agilent Infinium digital oscillo-scope are shown in Fig. 3.

The simulated results (Fig. 4.) and measured (Fig. 5) of the qualitatively different behavior of HR model in time domain are demonstrated. It can be see for xr = -0.6 system exhibits spiking behavior. If we change this bifurcation parameter to xr = -1.6 the system begins to exhibit chaotic behavior (chaotic dynamics is obtained for a small range around value xr = -1.6). With other change of xr is system exhibits bursting dynamics. It is evident that all the main dynamics of a neuron (spiking, bursting and chaos) can be obtained with the proposed circuit by properly setting the control parameters. It eventually turns out that this system in not as sensitive as expected.

Fig. 2. Simulated results in PSpice of the inertia neuron – plane projection.

4. Conclusions The HR neuron model has been implemented as a

fully analog circuit. We have shown through measurements that the proposed circuit can reproduce the main behavior of a biologically plausible neuron. The simulation results and measurements prove a good final agreement between theory and practice. The circuit synthesis of a single neuron can be the prelude to the implementation of neuron

networks. We point out that the integrator synthesis can be used to obtain circuit realizations of many nonlinear dy-namical systems, and it is not only limited to the HR model.

Fig. 3. Measured results of the inertia neuron – plane projection

and frequency spectrum.

Fig. 4. Simulated results of the qualitatively different behavior

of the HR model for a = 2.6, b = 4, d = 5, = 0.01, I = 2.99, xr = -0.6 (plot a), xr = -1.6 (plot b), xr = -2.0 (plot c), xr = -2.4 (plot d).

Fig. 5. Measured results of the qualitatively different behavior

of the HR model for a = 2.6, b = 4, d = 5, = 0.01, I = 2.99, xr = -0.6 (plot a), xr = -1.6 (plot b), xr = -2.0 (plot c), xr = -2.4 (plot d).

Acknowledgements The research leading to these results has received

funding from the European Community Seventh Frame-work Programme (FP7/2007-2013) under grant agreement no. 230126. Research described in the paper was also sup-ported by the Czech Ministry of Education under research program MSM 0021630513. The research is a part of the COST action IC 0803, which is financially supported by the Czech Ministry of Education under grant no. OC09016. The other author would also like to thank Grant Agency of the Czech Republic for their support through project num-ber 102/09/P217, 102/08/H027.

References [1] ZHAO, J., YONG-BIN, K. Circuit implementation of FitzHugh-

Nagumo neuron model using Field Programmable Analog Arrays. In Circuits and Systems, 2007. MWSCAS 2007. 50th Midwest Symposium. Montreal (Que., Canada), 2007, p. 772 - 775.

[2] BIZZARRI, F., LINARO, D., STORACE, M. PWL approximation of the Hindmarsh-Rose neuron model in view of its circuit implementation. In Circuit Theory and Design, 2007. ECCTD 2007. 18th European Conference. Seville (Spain), 2007, p. 878 – 881.

[3] HINDMARSH, J. L., ROSE, R. M. A model of neuronal bursting using three coupled first order differential equations. Proceedings of Royal Society B: Biological Sciences. London (England), 1984, p. 87-102.

[4] ITOH, M. Synthesis of electronic circuit for simulating nonlinear dynamics. International Journal of Bifurcation and Chaos, 2001, vol. 11, no. 3, p. 605-653. ISSN 0218-1274.

[5] POGGI, T., SCIUTTO, A., STORACE, M. Piecewise linear implementation of nonlinear dynamical systems: from theory to practice. Electronics Letters, 2009, vol. 45, no. 19, p. 966 – 967.

[6] STORACE, M., LINARO, D., LANGE, E. The Hindmarsh-Rose neuron model: bifurcation analysis and piecewiselinear approximations. Chaos, 2008, vol. 18, no. 3, p. 033128.

[7] PETRŽELA, J., HRUBOŠ, Z. Simplest chaos converters: modeling, analysis and future perspectives. Recent Advances in System Science and Simulation, WSEAS Transactions on Systems. Italy: WSEAS, 2009, p. 160-163. ISBN: 978-960-474-131- 1. ISSN: 1790- 2769.

[8] PETRŽELA, J., HRUBOŠ, Z. A note on chaos conversion in frequency domain. Recent Advances in Applied Mathematics, WSEAS Transactions on Systems. Canary Islands: WSEAS, 2009, p. 19-22. ISBN: 978-960-474-138- 0. ISSN: 1790- 2769.

[9] Analog Devices, Low Cost Analog Multiplier AD 633, 12 p. [Online] Available at: http://www.analog.com