Neural Spike Suppression by Adaptive Control of an Unknown Steady ...
Neural Spike Suppression by Adaptive Control of an Unknown Steady State Ar¯ unas Tamaˇseviˇcius1, Elena Tamaˇseviˇci¯ ut˙e1,2, Gytis Mykolaitis1,3 , 1 Skaidra Bumelien˙e , Raimundas Kirvaitis4 , and Ruedi Stoop5 1
Plasma Phenomena and Chaos Laboratory, Semiconductor Physics Institute, A. Goˇstauto 11, LT-01108 Vilnius, Lithuania [email protected], [email protected] http://www.pfi.lt/index_e 2 Department of General Physics and Spectroscopy, Faculty of Physics, Vilnius University, Saul˙etekio 9, LT-10222 Vilnius, Lithuania [email protected] http://www.vu.lt/en/ 3 Department of Physics, Faculty of Fundamental Sciences, Vilnius Gediminas Technical University, Saul˙etekio 11, LT-10223 Vilnius, Lithuania [email protected] http://www.fm.vgtu.lt/en/ 4 Department of Electronic Systems, Faculty of Electronics, Vilnius Gediminas Technical University, Naugarduko 41, LT-03227 Vilnius, Lithuania [email protected] http://www.el.vgtu.lt/en/ 5 Institute of Neuroinformatics, University of Z¨ urich (UNIZH) and Swiss Federal Institute of Technology Z¨ urich (ETHZ), Winterthurerstrasse 190, CH-8057 Z¨ urich, Switzerland [email protected] http://stoop.ini.uzh.ch
Abstract. A FitzHugh–Nagumo type spiking neuron model equipped with an asymmetric activation function is investigated. An analogue nonlinear electrical circuit imitating the dynamics of the model is proposed. It is demonstrated that a simple first order linear filter coupled to the system can inhibit spiking and stabilize the system on an unstable steady state, the position of which is not required to be known, since the filter operates as an adaptive controller. Analytical, numerical and experimental results are presented. Keywords: Adaptive control, Nonlinear dynamics, Neuron models.
1
Introduction
The problem of stabilizing unstable steady states (USS) is of great importance in nonlinear dynamics. Classical control methods require as a reference point the C. Alippi et al. (Eds.): ICANN 2009, Part I, LNCS 5768, pp. 618–627, 2009. c Springer-Verlag Berlin Heidelberg 2009
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coordinates of the USS. In many practical cases the location of the USS is either unknown or it may slowly vary with time. Therefore adaptive, reference-free methods, automatically locating the USS are preferable. The simplest adaptive technique for stabilizing USS is based on derivative controller. A perturbation in the form of a derivative dx/dt derived from an observable x(t) does not change the original system, since it vanishes when the variable approaches the steady state. This technique has been applied to stabilize a laser [1], an electrical circuit [2] and an electrochemical reaction [3]. Since the method requires differentiation it is rather sensitive to high frequency noise present in the signal x(t). Another adaptive method for stabilizing USS employs first order RC filters in the feedback loop [4,5,6,7,8,9,10]. Provided the cut-off frequency of the RC filter is low enough, the voltage accross the capacitor of the filter asymptotically approaches the USS and therefore can be used as a reference point for the proportional feedback. The method has been successfully applied to several experimental systems, including electrical circuits [4,5,6] and lasers [7,8]. Recently its modifications have been used to control unstable spirals in mathematical models, such as the Lorenz system [9,10], the van der Pol oscillator [10], and the parametrically excited mechanical pendulum [10]. Neuron models are dynamical systems exhibiting a rich variety of nonlinear phenomena, ranging from excitability, spiking, bursting, chaotic response to external periodic forcing, high-frequency inhibitory effects, to various spatio-temporal patterns in networks of coupled neurons. The FitzHugh–Nagumo (FHN) model [11] is one of the most popular models, due to its simplicity. Besides analytical and numerical studies of the different models, efforts have also concentrated on designing and building analogue electronic neurons. Replacement of biological neurons with electronic devices governed by simple equations, can contribute to a better understanding of the biological effects. A number of nonlinear electrical circuits imitating dynamical behaviour of neurons have been described in the literature. Among them are various modifications of the FHN electronic cells [12,13,14,15], the Hodgkin–Huxley (HH) neuron type [16], and the Liao’s time delayed [17] electronic neurons. Some earlier electronic implementations, e.g. of the Lewis and the Harman models, are collected in Ref. [18]. In the present paper, we describe an extremely simple modification of the FHN model by an asymmetric activation function, to which, for the sake of brevity, we will refer to as the asymmetric FHN (AFHN) model. We derive its experimental electronic analog and demonstrate analytically, numerically and experimentally that the USS can be robustly controlled by a simple first order filter.
2
Mathematical Model
We consider the folowing set of equations: x˙ = ax − fd (x) − y − ξ,
(1)
y˙ = x − by,
(2)
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⎧ ⎨ d1 (x + 1), x < −1 0, −1 ≤ x ≤ 1 fd (x) = ⎩ d2 (x − 1), x > 1 .
(3)
All coefficients in Eqs. (1–3) are positive and constant, except the parameter ξ, which, in general, will be unknown and/or may slowly vary with time. Though the coefficients d1 and d2 are somewhat arbitrary (the only requirement is d1,2 > a), we consider the case of strong asymmetry d2 1) values of x. For ab < 1,
|ξ| < 1/b − a
,
(4)
Eqs.(1–3) lead a single unstable steady state of coordinates x0 = −
bξ , 1 − ab
y0 = −
ξ 1 − ab
,
(5)
while there are no stable solutions. Thus, conditions (4) yield oscillatory solutions of Eqs. (1–2). Evidently, |x0 | < 1 and for ξ > 0 the steady state is negative x0 < 0, y0 < 0. When linearized around the steady state (x0 , y0 ), system (1–2) leads to the characteristic equation λ2 − (a − b)λ + 1 − ab = 0, which has two solutions that are independent on ξ: (a + b)2 a−b ± − 1. λ1,2 = 2 4
(6)
(7)
For a > b, the real parts of λ1,2 are positive, confirming that the steady state given by (5) is unstable (either an unstable spiral or an unstable node). When a > b and a + b > 2 both solutions are positive and real (no imaginary part). In this case, the steady state is an unstable node. Now we add to Eqs. (1–2) the third equation describing a stable RC filter and couple it to the AFHN equations: x˙ = ax − fd (x) − y − ξ − k(x − z), y˙ = x − by, z˙ = ωf (x − z),
(8) (9) (10)
where k is the coupling coefficient and ωf is the normalized threshold frequency of the filter. This system has the same steady state solution as the free-running system: bξ ξ , y0 = − , (11) x0 = z0 = − 1 − ab 1 − ab
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which implies that the filter does not influence the position of the steady state (x0 , y0 ) of the AFHN system, but can change its stability properties. The corresponding characteristic equation of the linearized system is λ3 + h3 λ2 + h2 λ + h1 = 0,
(12)
with h3 = −a + b + k + ωf , h2 = 1 − ab + bk − (a − b)ωf , h1 = (1 − ab)ωf . (13) The system is stable if the real parts of all three eigenvalues Reλ1,2,3 are negative. The results of the numerical solution of the characteristic equation (12) are shown in Fig. 1, in dependence of the coupling coefficient, for chosen system parameters a = 6, b = 0.1, ωf = 0.1. The largest eigenvalues Reλ cross zero and become negative at k = 5.9. The optimal values of the coupling coefficient kopt , providing the highest rate (λ = −0.1) of convergence to the stabilized steady state, are from k ≈ 6.5 to k = 10. The necessary and sufficient conditions for stabilization can be found using the Hurwitz matrix ⎞ ⎛ h3 h1 0 (14) H = ⎝ 1 h2 0 ⎠ . 0 h3 h1 According to the Routh-Hurwitz stability criterion the eigenvalues Reλ1,2,3 are all negative if the diagonal minors of the H matrix are all positive Δ1 = h3 > 0, Δ2 = h3 h2 − h1 > 0, Δ3 = h1 Δ2 > 0.
(15)
Since Δ2 should be positive according to the second inequality, the third inequality for Δ3 can be replaced simply with h1 > 0. This can be further simplified to (1 − ab) > 0 since ωf > 0 by definition. We note that, due to inequality (4), the last inequality will always be satisfied. Consequently, we are left with the
Fig. 1. Dependence of the Reλ on the coupling coefficient k from Eq. (12). a = 6, b = 0.1, ωf = 0.1. The insert is a zoomed view of the Reλ in the k range from 5 to 12.
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Fig. 2. Spikes x(t) from Eq. (1–3). a = 6, b = 0.1, ξ = 1.7, d1 = 60, d2 = 7.
Fig. 3. Main variable x(t) (top trace) and control signal x − z (bottom trace) in the case the control is switched on between two spikes.The control signal display is lowered for the sake of clarity. a = 6, b = 0.1, ξ = 1.7, d1 = 60, d2 = 7, ωf = 0.1, k = 9.
Fig. 4. Main variable x(t) (top trace) and control signal x − z (bottom trace) in the case the control is switched on during a spike. Parameters the same as in Fig. 3.
first and the second inequality of (15). We make an estimation of the threshold coupling coefficient kth by requiring that for k > kth , the two minors Δ1,2 should be positive. The result kth = 5.9 obtained for the system parameters is in good agreement with the corresponding value found from Reλ(k) in Fig. 1. Results of
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the numerical integration of Eqs. (1–3) as well as of Eqs. (8–10) are shown in Fig. 2, in Fig. 3, and in Fig. 4. The waveform in Fig. 2 generated by the AFHN model more closely resembles the behavior of the spiking neurons in the HH model than that of the classical FHN model with symmetric activation function. When the control (k > kth ) is turned on, the spikes are totally suppressed and the system is stabilized on non-zero steady state x0 , where the control signal x − z vanishes. The transient process is extremely short: after control is on, only one short spike emerges.
3
Analogue Circuit and Experimental Results
The electrical circuit imitating the dynamics of the AFHN system along with the RC adaptive controller is sketched Fig. 5. The element values of the electronic neuron cell were chosen as R1 = R2 = 1 kΩ, R3 = R6 = 200 Ω, R4 = 30 Ω, R5 = 240 Ω, R7 = 620 Ω, C = 330 nF, L = 1 H. The diodes D1 and D2 are the D1N5820 or similar type Schottky devices (the forward voltage drop V ∗ ≈ 0.2 V at 1 mA). The operational amplifier OA is the LM741 type IC. The inductor L was implemented as an active gyrator subcircuit [19]. The element values of the controller were R∗ = 200 Ω, C ∗ = 22 μF. Using the Kirchhoff’s laws the following equations for the circuit in Fig. 5 can be written: VC dVC VC V0 VC − VC ∗ = − − Id (VC ) − IL − − , dt R3 R7 R3 R∗ dIL = VC − R6 IL , L dt VC − VC ∗ dVC ∗ = , C∗ dt R∗ C
where
⎧ VC < −V ∗ ⎨ (VC + V ∗ )/R4 , 0, −V ∗ ≤ VC ≤ V ∗ Id (VC ) = ⎩ ∗ (VC − V )/R5 , VC > V ∗ .
(16) (17) (18)
(19)
VC
IL
VC*
Fig. 5. Circuit diagram of the electronic neuron cell with an adaptive controller. +V0 is a DC bias. The switch S, the resistor R* and the capacitor C* compose the controller.
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By introducing the following set of dimensionless variables and parameters ρ(R7 − R3 ) L VC ρIL t , ρ= , a= x = ∗, y = ∗ , t = √ , V V C R3 R7 LC R6 ρV0 ρ ρ ρ ρC b= , ξ= , d1 = , d2 = , k = ∗ , ωf = ∗ ∗ . ρ R3 V ∗ R4 R5 R R C Here R∗ is chosen to provide appropriate value of k, while C ∗ is chosen to ensure low enough value of the ωf . Eqs. (16–18) can be presented in the form exactly coinciding with Eqs. (8–10), while Eq. (19) transforms into Eq. (3). Corresponding experimental results are presented in the Figs. 6–9. The spike train (Fig. 7) and the controlled dynamics (Figs. 8,9) are in a good agreement with the numerical results, presented in the prvious section. I, mA 1
-1
-0.5
0
0.5
1 V, V
-1
-2
Fig. 6. DC Current-voltage characteristic of the electronic neuron cell. The elements C, L and R6 are removed from the circuit, the switch S is in the open position.
1V
0 5 ms Fig. 7. Typical train of spikes VC (t) from the circuit without control
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1V
0
0.1 V
5 ms
Fig. 8. Output voltage VC (t) (top trace) and control signal VR∗ (bottom trace) if the control is switched on during the interspike-interval. The control signal display is lowered for the sake of clarity.
1V 0
1V 5 ms Fig. 9. Output voltage VC (t) (top trace) and control signal VR∗ (bottom trace) if the control is switched on during a spike. The control signal display is lowered for the sake of clarity.
4
Conclusions
In this work, a modification of the classical FitzHugh–Nagumo (FHN) model is proposed. The common symmetric activation function x − x3 is replaced with a three-segment piecewise linear asymmetric function. In contrast to the common symmetric model at standard parameter values, the asymmetric version of the FHN model (AFHN) exhibits narrow triangular shaped spikes, similarly to more complicated biologically inspired Hodgkin–Huxley model. Besides, a nonlinear
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electronic analog has been designed, built and investigated experimentally. Besides, we have applied the control method, based on the linear first order RC filter, to stabilize the unstable steady state of a spiking neuron. The method is fully adaptive; it automatically locates the steady state the coordinates of which can be left unevaluated. Acknowledgments. The work was supported in part by the Lithuanian State Science and Studies Foundation.
References 1. Bielawski, S., Bouazaoui, M., Derozier, D., Glorieux, P.: Stabilization and Characterization of Unstable Steady State in a Laser. Phys. Rev. A 47, 3276–3279 (1993) 2. Johnson, G.A., Hunt, E.R.: Derivative Control of the Steady State in Chua’s Circuit Driven in the Chaotic Region. IEEE Trans. Circuit Syst. 40, 833–834 (1993) 3. Parmananda, P., Rhode, M.A., Johnson, G.A., Rollins, R.W., Dewald, H.D., Markworth, A.J.: Stabilization of Unstable Steady Sstate in an Electrochemical System Using Derivative Control. Phys. Rev. E 49, 5007–5011 (1994) 4. Rulkov, N.F., Tsimring, L.S., Abarbanel, H.D.I.: Tracking Unstable Orbits in Chaos Using Dissipative Feedback Control. Phys. Rev. E 50, 314–324 (1994) 5. Namaj¯ unas, A., Pyragas, K., Tamaˇseviˇcius, A.: Stabilization of an Unstable Steady State in a Mackey-Glass System. Phys. Lett. A 204, 255–262 (1995) 6. Namaj¯ unas, A., Pyragas, K., Tamaˇseviˇcius, A.: Analog Techniques for Modeling and Controlling the Mackey-Glass System. Int. J. Bif. Chaos 7, 957–962 (1997) 7. Ciofini, M., Labate, A., Meucci, R., Galanti, M.: Stabilization of Unstable Fixed Points in the Dynamics of a Laser with Feedback. Phys. Rev. E 60, 398–402 (1999) 8. Schenck zu Schweinsberg, A., Dressler, U.: Characterization and Stabilization of the Unstable Fixed Points of a Frequency Doubled Nd: YAG Laser. Phys. Rev. E 63, 056210 (2001) 9. Huijberts, H.: Linear Controllers for the Stabilization of Unknown Steady States of Chaotic systems. IEEE Trans. Circuits and Syst.–I 53, 2246–2254 (2006) 10. Brown, D.J.: Adaptive Steady-State Stabilization for Nonlinear Dynamical Systems. Phys. Rev. E 78, 16213 (2008) 11. Roc¸soreanu, C., Georgescu, A., Giugit¸eanu, N.: The FitzHugh-Nagumo Model: Bifurcation and Dynamics. Kluwer Academic Publishers, Dordrecht (2000) 12. Binczak, S., Kazantsev, V.B., Nekorkin, V.I., Bilbaut, J.M.: Experimental Study of Bifurcations in Modified FitzHugh-Nagumo Cell. Electron. Lett. 39, 961–962 (2003) 13. Aliaga, J., Busca, N., Minces, V., Mindlin, G.B., Pando, B., Salles, A., Sczupak, L.: Electronic Neuron within a Ganglion of a Leech (Hirudo Medicinalis). Phys. Rev. E 67, 061915 (2003) 14. Jacquir, S., Binczak, S., Bilbaut, J.M., Kazantsev, V.B., Nekorkin, V.I.: Study of Electronic Master-Slave MFHN Neurons. In: 12th International Workshop on Nonlinear Dynamics of Electronic Systems, pp. 182–185. Centro de Geofisica de ´ ´ ´ Evora, Universidade de Evora, Evora (2004) 15. Jacquir, S., Binczak, S., Bilbaut, J.-M., Kazantsev, V., Nekorkin, V.: Synaptic Coupling Between Two Electronic Neurons. Nonl. Dyn. 44, 29–36 (2006)
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16. Sitt, J.D., Aliaga, J.: Versatile Biologically Inspired Electronic Neuron. Phys. Rev. E 76, 051919 (2007) 17. Duan, S., Liao, X.: An Electronic Implementation for Liao’s Chaotic Delayed Neuron Model with Non-Monotonous Activation Function. Phys. Lett. A 369, 37–43 (2007) 18. Malmivuo, J., Plonsey, R.: Bioelectromagnetism – Principles and Applications of Bioelectric and Biomagnetic Fields. Oxford University Press, New York (1995), http://www.bem.fi/book/ 19. Horowitz, P., Hill, W.: The Art of Electronics. Cambridge University Press, Cambridge (1993)
Plasma Phenomena and Chaos Laboratory, Semiconductor Physics Institute, A. Goˇstauto 11, LT-01108 Vilnius, Lithuania [email protected], [email protected] http://www.pfi.lt/index_e 2 Department of General Physics and Spectroscopy, Faculty of Physics, Vilnius University, Saul˙etekio 9, LT-10222 Vilnius, Lithuania [email protected] http://www.vu.lt/en/ 3 Department of Physics, Faculty of Fundamental Sciences, Vilnius Gediminas Technical University, Saul˙etekio 11, LT-10223 Vilnius, Lithuania [email protected] http://www.fm.vgtu.lt/en/ 4 Department of Electronic Systems, Faculty of Electronics, Vilnius Gediminas Technical University, Naugarduko 41, LT-03227 Vilnius, Lithuania [email protected] http://www.el.vgtu.lt/en/ 5 Institute of Neuroinformatics, University of Z¨ urich (UNIZH) and Swiss Federal Institute of Technology Z¨ urich (ETHZ), Winterthurerstrasse 190, CH-8057 Z¨ urich, Switzerland [email protected] http://stoop.ini.uzh.ch
Abstract. A FitzHugh–Nagumo type spiking neuron model equipped with an asymmetric activation function is investigated. An analogue nonlinear electrical circuit imitating the dynamics of the model is proposed. It is demonstrated that a simple first order linear filter coupled to the system can inhibit spiking and stabilize the system on an unstable steady state, the position of which is not required to be known, since the filter operates as an adaptive controller. Analytical, numerical and experimental results are presented. Keywords: Adaptive control, Nonlinear dynamics, Neuron models.
1
Introduction
The problem of stabilizing unstable steady states (USS) is of great importance in nonlinear dynamics. Classical control methods require as a reference point the C. Alippi et al. (Eds.): ICANN 2009, Part I, LNCS 5768, pp. 618–627, 2009. c Springer-Verlag Berlin Heidelberg 2009
Neural Spike Suppression
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coordinates of the USS. In many practical cases the location of the USS is either unknown or it may slowly vary with time. Therefore adaptive, reference-free methods, automatically locating the USS are preferable. The simplest adaptive technique for stabilizing USS is based on derivative controller. A perturbation in the form of a derivative dx/dt derived from an observable x(t) does not change the original system, since it vanishes when the variable approaches the steady state. This technique has been applied to stabilize a laser [1], an electrical circuit [2] and an electrochemical reaction [3]. Since the method requires differentiation it is rather sensitive to high frequency noise present in the signal x(t). Another adaptive method for stabilizing USS employs first order RC filters in the feedback loop [4,5,6,7,8,9,10]. Provided the cut-off frequency of the RC filter is low enough, the voltage accross the capacitor of the filter asymptotically approaches the USS and therefore can be used as a reference point for the proportional feedback. The method has been successfully applied to several experimental systems, including electrical circuits [4,5,6] and lasers [7,8]. Recently its modifications have been used to control unstable spirals in mathematical models, such as the Lorenz system [9,10], the van der Pol oscillator [10], and the parametrically excited mechanical pendulum [10]. Neuron models are dynamical systems exhibiting a rich variety of nonlinear phenomena, ranging from excitability, spiking, bursting, chaotic response to external periodic forcing, high-frequency inhibitory effects, to various spatio-temporal patterns in networks of coupled neurons. The FitzHugh–Nagumo (FHN) model [11] is one of the most popular models, due to its simplicity. Besides analytical and numerical studies of the different models, efforts have also concentrated on designing and building analogue electronic neurons. Replacement of biological neurons with electronic devices governed by simple equations, can contribute to a better understanding of the biological effects. A number of nonlinear electrical circuits imitating dynamical behaviour of neurons have been described in the literature. Among them are various modifications of the FHN electronic cells [12,13,14,15], the Hodgkin–Huxley (HH) neuron type [16], and the Liao’s time delayed [17] electronic neurons. Some earlier electronic implementations, e.g. of the Lewis and the Harman models, are collected in Ref. [18]. In the present paper, we describe an extremely simple modification of the FHN model by an asymmetric activation function, to which, for the sake of brevity, we will refer to as the asymmetric FHN (AFHN) model. We derive its experimental electronic analog and demonstrate analytically, numerically and experimentally that the USS can be robustly controlled by a simple first order filter.
2
Mathematical Model
We consider the folowing set of equations: x˙ = ax − fd (x) − y − ξ,
(1)
y˙ = x − by,
(2)
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⎧ ⎨ d1 (x + 1), x < −1 0, −1 ≤ x ≤ 1 fd (x) = ⎩ d2 (x − 1), x > 1 .
(3)
All coefficients in Eqs. (1–3) are positive and constant, except the parameter ξ, which, in general, will be unknown and/or may slowly vary with time. Though the coefficients d1 and d2 are somewhat arbitrary (the only requirement is d1,2 > a), we consider the case of strong asymmetry d2 1) values of x. For ab < 1,
|ξ| < 1/b − a
,
(4)
Eqs.(1–3) lead a single unstable steady state of coordinates x0 = −
bξ , 1 − ab
y0 = −
ξ 1 − ab
,
(5)
while there are no stable solutions. Thus, conditions (4) yield oscillatory solutions of Eqs. (1–2). Evidently, |x0 | < 1 and for ξ > 0 the steady state is negative x0 < 0, y0 < 0. When linearized around the steady state (x0 , y0 ), system (1–2) leads to the characteristic equation λ2 − (a − b)λ + 1 − ab = 0, which has two solutions that are independent on ξ: (a + b)2 a−b ± − 1. λ1,2 = 2 4
(6)
(7)
For a > b, the real parts of λ1,2 are positive, confirming that the steady state given by (5) is unstable (either an unstable spiral or an unstable node). When a > b and a + b > 2 both solutions are positive and real (no imaginary part). In this case, the steady state is an unstable node. Now we add to Eqs. (1–2) the third equation describing a stable RC filter and couple it to the AFHN equations: x˙ = ax − fd (x) − y − ξ − k(x − z), y˙ = x − by, z˙ = ωf (x − z),
(8) (9) (10)
where k is the coupling coefficient and ωf is the normalized threshold frequency of the filter. This system has the same steady state solution as the free-running system: bξ ξ , y0 = − , (11) x0 = z0 = − 1 − ab 1 − ab
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which implies that the filter does not influence the position of the steady state (x0 , y0 ) of the AFHN system, but can change its stability properties. The corresponding characteristic equation of the linearized system is λ3 + h3 λ2 + h2 λ + h1 = 0,
(12)
with h3 = −a + b + k + ωf , h2 = 1 − ab + bk − (a − b)ωf , h1 = (1 − ab)ωf . (13) The system is stable if the real parts of all three eigenvalues Reλ1,2,3 are negative. The results of the numerical solution of the characteristic equation (12) are shown in Fig. 1, in dependence of the coupling coefficient, for chosen system parameters a = 6, b = 0.1, ωf = 0.1. The largest eigenvalues Reλ cross zero and become negative at k = 5.9. The optimal values of the coupling coefficient kopt , providing the highest rate (λ = −0.1) of convergence to the stabilized steady state, are from k ≈ 6.5 to k = 10. The necessary and sufficient conditions for stabilization can be found using the Hurwitz matrix ⎞ ⎛ h3 h1 0 (14) H = ⎝ 1 h2 0 ⎠ . 0 h3 h1 According to the Routh-Hurwitz stability criterion the eigenvalues Reλ1,2,3 are all negative if the diagonal minors of the H matrix are all positive Δ1 = h3 > 0, Δ2 = h3 h2 − h1 > 0, Δ3 = h1 Δ2 > 0.
(15)
Since Δ2 should be positive according to the second inequality, the third inequality for Δ3 can be replaced simply with h1 > 0. This can be further simplified to (1 − ab) > 0 since ωf > 0 by definition. We note that, due to inequality (4), the last inequality will always be satisfied. Consequently, we are left with the
Fig. 1. Dependence of the Reλ on the coupling coefficient k from Eq. (12). a = 6, b = 0.1, ωf = 0.1. The insert is a zoomed view of the Reλ in the k range from 5 to 12.
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Fig. 2. Spikes x(t) from Eq. (1–3). a = 6, b = 0.1, ξ = 1.7, d1 = 60, d2 = 7.
Fig. 3. Main variable x(t) (top trace) and control signal x − z (bottom trace) in the case the control is switched on between two spikes.The control signal display is lowered for the sake of clarity. a = 6, b = 0.1, ξ = 1.7, d1 = 60, d2 = 7, ωf = 0.1, k = 9.
Fig. 4. Main variable x(t) (top trace) and control signal x − z (bottom trace) in the case the control is switched on during a spike. Parameters the same as in Fig. 3.
first and the second inequality of (15). We make an estimation of the threshold coupling coefficient kth by requiring that for k > kth , the two minors Δ1,2 should be positive. The result kth = 5.9 obtained for the system parameters is in good agreement with the corresponding value found from Reλ(k) in Fig. 1. Results of
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the numerical integration of Eqs. (1–3) as well as of Eqs. (8–10) are shown in Fig. 2, in Fig. 3, and in Fig. 4. The waveform in Fig. 2 generated by the AFHN model more closely resembles the behavior of the spiking neurons in the HH model than that of the classical FHN model with symmetric activation function. When the control (k > kth ) is turned on, the spikes are totally suppressed and the system is stabilized on non-zero steady state x0 , where the control signal x − z vanishes. The transient process is extremely short: after control is on, only one short spike emerges.
3
Analogue Circuit and Experimental Results
The electrical circuit imitating the dynamics of the AFHN system along with the RC adaptive controller is sketched Fig. 5. The element values of the electronic neuron cell were chosen as R1 = R2 = 1 kΩ, R3 = R6 = 200 Ω, R4 = 30 Ω, R5 = 240 Ω, R7 = 620 Ω, C = 330 nF, L = 1 H. The diodes D1 and D2 are the D1N5820 or similar type Schottky devices (the forward voltage drop V ∗ ≈ 0.2 V at 1 mA). The operational amplifier OA is the LM741 type IC. The inductor L was implemented as an active gyrator subcircuit [19]. The element values of the controller were R∗ = 200 Ω, C ∗ = 22 μF. Using the Kirchhoff’s laws the following equations for the circuit in Fig. 5 can be written: VC dVC VC V0 VC − VC ∗ = − − Id (VC ) − IL − − , dt R3 R7 R3 R∗ dIL = VC − R6 IL , L dt VC − VC ∗ dVC ∗ = , C∗ dt R∗ C
where
⎧ VC < −V ∗ ⎨ (VC + V ∗ )/R4 , 0, −V ∗ ≤ VC ≤ V ∗ Id (VC ) = ⎩ ∗ (VC − V )/R5 , VC > V ∗ .
(16) (17) (18)
(19)
VC
IL
VC*
Fig. 5. Circuit diagram of the electronic neuron cell with an adaptive controller. +V0 is a DC bias. The switch S, the resistor R* and the capacitor C* compose the controller.
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By introducing the following set of dimensionless variables and parameters ρ(R7 − R3 ) L VC ρIL t , ρ= , a= x = ∗, y = ∗ , t = √ , V V C R3 R7 LC R6 ρV0 ρ ρ ρ ρC b= , ξ= , d1 = , d2 = , k = ∗ , ωf = ∗ ∗ . ρ R3 V ∗ R4 R5 R R C Here R∗ is chosen to provide appropriate value of k, while C ∗ is chosen to ensure low enough value of the ωf . Eqs. (16–18) can be presented in the form exactly coinciding with Eqs. (8–10), while Eq. (19) transforms into Eq. (3). Corresponding experimental results are presented in the Figs. 6–9. The spike train (Fig. 7) and the controlled dynamics (Figs. 8,9) are in a good agreement with the numerical results, presented in the prvious section. I, mA 1
-1
-0.5
0
0.5
1 V, V
-1
-2
Fig. 6. DC Current-voltage characteristic of the electronic neuron cell. The elements C, L and R6 are removed from the circuit, the switch S is in the open position.
1V
0 5 ms Fig. 7. Typical train of spikes VC (t) from the circuit without control
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1V
0
0.1 V
5 ms
Fig. 8. Output voltage VC (t) (top trace) and control signal VR∗ (bottom trace) if the control is switched on during the interspike-interval. The control signal display is lowered for the sake of clarity.
1V 0
1V 5 ms Fig. 9. Output voltage VC (t) (top trace) and control signal VR∗ (bottom trace) if the control is switched on during a spike. The control signal display is lowered for the sake of clarity.
4
Conclusions
In this work, a modification of the classical FitzHugh–Nagumo (FHN) model is proposed. The common symmetric activation function x − x3 is replaced with a three-segment piecewise linear asymmetric function. In contrast to the common symmetric model at standard parameter values, the asymmetric version of the FHN model (AFHN) exhibits narrow triangular shaped spikes, similarly to more complicated biologically inspired Hodgkin–Huxley model. Besides, a nonlinear
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electronic analog has been designed, built and investigated experimentally. Besides, we have applied the control method, based on the linear first order RC filter, to stabilize the unstable steady state of a spiking neuron. The method is fully adaptive; it automatically locates the steady state the coordinates of which can be left unevaluated. Acknowledgments. The work was supported in part by the Lithuanian State Science and Studies Foundation.
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