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Neural modeling with dynamically adjustable threshold and refractory period
BioSystems, 27 (1992) 137-144
137
Elsevier Scientific Publishers Ireland Ltd.
Neural modeling with dynamically adjustable threshold and refractory period Qiang Gan and YU Wei Department of BiomedicaJ:Engineering, Southeast University, Nanfing ~10018 (P.R. of China) (Received December 12th,~ 1991) (Revision received August llth, 1992)
A variant of the FitzHugh-Nagumo model is proposed in order to fully make use of the computational properties of intraneuronal dynamics. The mechanisms of threshold and refractory periods resulting from the double dynamical processes are qualitatively studied through computer simulation. The results show that the variant neuron model has the property that its threshold, refractory period and response amplitude are dynamically adjustable. This paper has also discussed some problems relating to collective property, learning and implementation of the neural network based on the neuron model proposed. It is noted that the implicit way to describe threshold and refractory period is advantageous to adaptive learning in neural networks and that molecular electronics probably provides an effective approach to implementing the above neuron model.
Keywo~ds: Neuron model; Refractory period; Threshold dynamics; Neural networks.
1. Introduction
Mathematical modeling for neural systems aims at understanding the mechanisms of biological neural systems and providing the basis for artificial neural network modeling. The first mathematical neuron model was proposed and applied to construct neural nets based on simple logic calculus in 1943 (McCulloch and Pitts, 1943), which have played a very important role in promoting studies in information science, automation and computer systems' theory and is often called the M-P model. In 1952, Hodgkin and Huxley put for~yard a set of equations to describe the excitation and conduction of neural impulses in the squid axon (Hodgkin and Huxley, 1952), which is still the most widely accepted standard mathematical model for biological neurons. The Hodgkin-Huxley equaCarrespondenee to: Qiang Gan, Department of Biomedical Engineering, Southeast University, Nanjing 210018, P.R. of China.
tions (H-H equations) constitute a very complicated four-dimensional dynamical system. Either from the viewpoint of applications or from that of mathematical analysis, it is quite necessary to simplify the equations while mainraining the basic computational properties. Since then there has been a lot of research on the simplification and improvement of the H-H equations (MacGregor and Lewis, 1976), among them the FitzHugh-Nagumo model (FitzHugh, 1961), with the simplest mathematical expression, is the most widely acceptable simplified model of the H-H equations and is highly recommended for its potential applications in artificial neural network modeling (Sanchez-Sinencio, 1991). In recent years, scientists have turned much of their attention to the architecture of interconnection between neurons, but a little to the neuronal modeling itself. However, more and more evidence shows that intraneuronal dynamics contribute a great deal to collective properties of neural nets (e.g. Horn and Usher, 1989; Yanai, 1990).
0303-2647/92/$05.00 © 1992 Elsevier ScientificPublishers Ireland Ltd. Printed and Published in Ireland
138 The non-linear properties such as threshold and refractory periods play a key role in the information processing of biological neural systems. However, the M-P model has been used in most artificial neural networks, in which only the time-invariant threshold property is considered. It is shown that introducing a timevariant threshold function into the neural network model allows for the improvement of the information storage capacity and associative ability of the neural network (Horn and Usher, 1989; Yanal, 1990). In fact, the refractory period is equivalent to the time-variant threshold effect, which is advantageous to improving the performance of neural networks. The authors think that the refractory period expressed in an implicit (or distributed) way is more advantageous to the adaptive learning in neural nets and is also in accordance with the information processing mode in biological neural systems. In this respect, the FitzHugh-Nagumo model would be considered a good candidate (Sanchez-Sinencio, 1991), but the dynamical range of the refractory period in this model is very much limited. Especially, when the refractory period is shortened to some extent by adjusting certain parameters in the model, the threshold effect will no longer exist. On the other hand, in the practical application of neural networks different tasks often require different ranges of refractory periods (Horn and Usher, 1989; Yanai, 1990). Our paper presents a variant of the FitzHugh-Nagumo model, in which the dynamical range of threshold, refractory period and response amplitude can be arbitrarily adjusted and satisfy practical requirements.
2. Qualitative analysis and mathematical description So far the H-H equations are the most satisfactory mathematical model for biological neurons. The FitzHugh-Nagumo model, a two~limensional dynamical system, is the simplest continuous mathematical model, basically preserving the computational properties of the H-H equations. The model is described by the
following equations:d y / d t ffi - ( x - a + by)/c
(1)
d~ldt ffi c(y + x - X3/3 + Z)
(2)
where x and y are dynamical variables, z represents the stimulus and a, b and c are control parameters. Our computer simulation analysis shows that in order to produce threshold and refractory effect, the variables x and y should interact with each other and their change rates have to be considerably different. In addition, if one variable represents the excitation in response to the stimulus, the other has to represent the inhibition (or accommodation). The FitzHughNagumo model satisfies the above conditions, but remains somewhat deficient. Although the refractory period can be adjusted by varying the parameter c in the model, its dynamical range is very much limited. Because c controls the speeds of two dynamical processes, when c is near to or less than one, the speed difference of the two dynamical processes is not obvious. As a result, the threshold effect disappears in the FitzHugh-Nagumo model. This is shown in Fig. 1, which includes the x , y trajectory and two lines (x-nullcline and y-nullcline) representing d x / d t = 0 a n d d y / d t = O, respectively. In Fig. l(a) the stimulus is too weak to cross the xnullcline. In Fig. l(b) the stimulus is strong enough to cross the x-nullcline. If a threshold effect exists, the response should appear to be a sudden increase when the stimulus is strong enough to cross the x-nullcline. However, because c equals 1 in Fig. 1, there is no such sudden increase in the response to the stimulus, no matter whether the stimulus is weak or strong. The computer simulation shows that the shortest refractory period (including relative refractory period) of the FitzHugh-Nagumo model is around 5 ms.. As a matter of fact, one of a general biological neuron is 1 or 2 ms. and in the artificial neural network study a large dynamical range of refractory period could be expected (Horn and Usher, 1989). In addition, the response amplitude in the FitzHugh-
139
Parameters: a=-I b=2 c=1
Y 1
/
d y / d t = O: / = (a - x)/b
f
d x / d t = O: y=x,x,x/3-x
-1
Please input pulse amplitude: Z = 1 Please input the interval between pulses (ms): invl = 20 Please input the number of pulses: count = 1 Press space key to exit I Press Y to change constants 1 Press any other keys to continue I
Parameters: a=-I b=2 c : 1
-'~
d y / d t = O: y = (a - x)/b dx/dt = O: y=x,x,x/3-x
"
ly
(//i
b
x
-1L
Please input pulse amplitude: Z = 1.5 Please input the interval between pulses (ms): invl = 20 Please input the number of pulses: count = 1 Press space key to exit I Press Y to change constants I Press any other keys to continue I
Fig. 1. Effect of parameter c in Fizhugh-Nagumo model on threshold. (a) S t h n u h s is too weak to cross x-nullcline; (b) stimulus is strong enough to cross x-nullcline. In both cases there is no sudden change in response after stimulation because c ffi 1.
Nagumo model mainly depends on the shape of the x-nullcline, which is fixed so that the response amplitude iis basically unadjustable. Based on the above analysis, we propose the following differential equations as a variant in order to overcome the deficiency of the FitzHugh-Nagumo nlodel: dy/dt ffi -0.5 M (At
+
A2x + Y)
(3)
d~/dt = 2 M(x
-
Bx 3
+
Cy) + Ds
(4)
where x and y represent excited and inhibited variables respectively, s represents the stimulus and A1, A2, B, C, D, M are control parameters. In the FitzHugh-Nagumo model, the threshold effect is emphasized, while the refractory period is somewhat neglected. In our variant model, the refractoriness and its linear adjustability are
140 two of the most important properties of the neuron model. The variation of the variables x and y depends mainly on the shape and position of the nullclines which are described by the following equations: y =
-
(A 1 + A2X )
(5)
- Bx3)/C
(6)
y = - (x
The crossing point of the nullclines is the equilibrium point of the system. Obviously, parameter -A1 is the intercept of the y-nullcline at the y axis, A2 controls its slope. From Eqns. 3 and 4 we also know that the sign reversal of A1 changes the sign of x, y and s to the reverse. Parameters B and C affect the shape of the xnullcline so that the amplitudes of x and y can be controlled, where C only controls the dilation of the x-nullcline in the y direction and B mainly controls the dilation in the x direction. It is easy to find out from Eqns. 3 and 4 that parameter M controls the change rates of x and y, thus controlling the duration of the refractoriness and no matter what the value of M is, the difference between the change rates of x and y is large enough to give rise to the threshold effect. In addition, the dynamical range of the threshold can be adjusted by parameter D. In the following section, more detailed results are given through computer simulation analysis. 3. Computer simulation analysis Because of the non-linearity in the model, the variation rule of x and y is very complicated and it is quite difficult to get the precise solution to Eqns. 3 and 4. Based on computer simulation, we can analyze the stimulus-response relationship, represented by the carves of x and y against time in response to the stimuli or the x , y trajectory in the x-y plane. Therefore, we can analyze the threshold and refractory effect of the two-dimensional dynamical system described by Eqns. 3 and 4. Especially, the influence of parameters in Eqns. 3 and 4 on threshold and refractory period can be analyzed in this way.
In the study of the curves of x and y against time, emphasis is placed on the responses to two adjacent impulse stimuli. The change rates of x and y, the duration of refractoriness and the influence of the parameters can be observed clearly from these curves. In the x - y plane, we mainly pay attention to the effect of the parameters on the shape and position of the nullclines. Therefore, it is easy to observe the equilibrium point and threshold effect in the x - y plane. With a set of typical parameter values, the curves of x and y against time and the trajectory in the x-y plane are shown in Figs. 2(a) and (b), respectively. In Fig. 2(a) the curve of x is plotted in a slender line and y in a thick line. In Fig. 2(b) the nullclines are plotted in slender line and the z,y trajectory in thick line. In the following figures, the convention about the slender line and thick line is the same as that in Fig. 2. In addition, the parameter values, stimulus strength and interval between two adjacent impulse stimuli are also given in Fig. 2. The typical parameter values are as follows: A1 = 0.5, A2-- 1, B = 1, C-- 0.5, D = 1, a n d M = 1. As shown in the following figures, some parameter values are changed and the results are compared with those from the typical parameter values so that the effect of these parameters on the performance of the model can be analyzed. Figure 3 demonstrates the effect of parameters A1 and A2 on the equilibrium point. When the values of A1 and A2 are changed from their typical values to 0.25 and 0.5, respectively, the position of the y-nullcline changes considerably. As a result, three equilibrium points exist in the system. When the system is stimulated at a certain equilibrium point, it probably becomes stable again at another equilibrium point. As shown in Fig. 4, parameter B mainly affects the shape of the x-nullcline, thus the amplitudes of x and y and the duration of refractoriness, when B increases, the shape of the xnullcline is compressed in both x and y directions, otherwise it is stretched. However, the dilation in the x direction is much more obvious than that in the y direction. In addition, by
......
141
Do you test pulse stimulation (y/n)? y Parameters: A 1 = 0.5 A 2 = 1 B= 1 C=0.5 D=I M=I
't x
x__
Z
W
1 0
dx/dt = 2M (x- B * x , x,x+C,y+D,S)
-1 dy/dt =
- 0 . 5 M (A1 x + y)
+
A2 * 0
8
16
t(ms)
|
I
24
32
.:1
2
Please input pulse amplitude: s = 0.8 Please input the time of next pulses (ms): T = 12 Press space key to exit I Press Y to change constants I Press any other keys to continue !
Parameters:
Y
A 1 = 0.5
A 2 -- 1
B= 1 D=I
C=0.5 M=I
dx/dt = 2M(x B,x,x,x+C,y +D,s)
t
-2
;
X
/"
dy/dt =
-0.SM(A 1 + A 2 * x+
y)
Please input pulse amplitude: s = 0.8 Please input the interval between pulses (ms): invl = 20 Please input the number of pulses: count = 1 Press space key to exit ! Press Y to change constants t Press any other keys to continue I
Fig. 2. Sfimulus-respon~,;e relationship of the improved model with a set of typical parameter values. (a) Curves of z and y against time; (b) trajectory in z ~ plane.
142
Parameters: A 1 = 0.25 A 2 = 0.5 B= 1 C=0.5 D=I M=I
Y
iI !I
ii
dx/dt = 2 M ( x B,x,x,x+C,y+ D,s)
i
X
2
dy/dt = - 0 . 5 M (,41 + A 2 , x+y)
Please input pulse amplitude: s = - 0 . 5 Please input the interval between pulses(ms): invI = 20 Please input the number of pulses: count = 1 Press space key to exit I Press Y to change constants i Press any other keys to countine I
Fig. 3. Effect of parameters A 1 and A 2 on equilibrium point and threshold.
Param~ers: A1=0.5 A2=1 B=3 C=0.5 D=I M=I
Y
,I
dx/dt = 2M(x B,x,x,x+C, +D,s)
dy/dt= - 0 5M(A1 + A2 =
y
I
-2
X
2
/
,L\
x+y)
Please input pulse amplitude: s = 0.8 Please input the interval between pulses (ms): invl = 20 Please input the number of pulses: count = 1 Press space key to exit I Press Y to change constants ! Press any other keys to countine !
Fig. 4. Effect of parameter B on shape of z-nullcline. When B increases, x-nullcline is compressed so that response amplitude descends.
143
carefully analyzing Eqns. 3 and 4, we can discover that when B is too small the shape of xnullcline is dominated by the linear term in Eqn. 6 so that the model can no longer simulate the properties of neurons. The effect of parameter C is very obvious, observed directly from Eqn. 6. When C increases, the x-nullcline is compressed only in the y direction, otherwise it is stretched. Therefore, the amplitudes of the x-nullcline in the x and y axes can be arbitrarily adjusted by combining the variation of B and C. The effect of parameter M can be clearly observed from the curves of x and y against time, as shown in Fig. 5. The parameter values are also listed in the figure. Parameter M only affects the change rates of x and y and thus the duration of the refractoriness but exerts no influence on the amplitudes of x and y and the threshold. The duration of refractoriness can be linearly adjusted by varying parameter M. When M increases, the refractory period is linearly shortened, otherwise it is extended.
The above analysis shows that the parameters in Eqns. 3 and 4 have obvious impact on the performance of the model. The parameter values can be adjusted to make the model satisfy any given targets of performances such as threshold and refractoriness, but it is impossible to do so in the FitzHugh-Nagumo model. For example, parameters B and C are firstly adjusted to guarantee the amplitudes of x and y. Next, A1 and A2 are adjusted to select a suitable equilibrium point. Then, D is adjusted to select any given threshold value. Finally, M can be adjusted to select a required refractory period. 4. Discussion and conclusion
Based on the above analysis, we think that the essence of the refractory period in neurons lies in the interaction of double (or multiple) dynamical processes with different speeds, which is in accordance with the viewpoint of neurobiology. Therefore, in order to obtain the refractory property in artificial neurons it is
Do you test pulse stimulus (y/n)? y
Parameters: : A 1 = 0.5
A2 = 1
B=I
C=0.5
D=I
M=4
x
_y
1'6
/(ms)
dx/dt = 2M(xB,x,x,x+C,y +D,s) dy/dt = - 0 . 5 M (A1 + A2 * x + y)
-2
1
0
8
24
312
Please input pulse amplitude: s = 0.8 Please input the time of next pulses (ms): T = 3 Press space key to exit ! Press Y to change constants I Press any other keys to continue I
Fig. 5. Effect of parameter M on refractory period. Refractory period can be arbitrarily adjusted by varying M. When M increases, duration of refractoriness is linearly shortened, otherwise it is extended.
144
necessary to introduce internal dynamics into neuron models, that is, there should be at least one internal variable to connect the stimulus to the response. A neuron model with dynamically adjustable threshold, refractory period and response amplitude is proposed in this paper, which could be used in artificial neural networks. It is shown that the refractory period (or time-variant threshold effect) plays a very important role in improving the collective properties of neural networks (Gan et al., 1990; Horn and Usher, 1989; Yanai, 1990). However, there still is a lack of sound theoretical analysis and wide experimental findings, thus leaving much to be desired in promoting the application of refractory property in artificial neural networks. Because the refractory property is essentially characteristic of the time domain, it is profitable to make use of the refractory period in pulseencoding neural network architecture. In addition, this kind of architecture has the features of saving power and space and combining the merits of digital and analogue signal processing (Meader et al., 1991; Murray et al., 1991). So far there are still some problems to be solved for learning in the neural network of this architecture. It was noted that the reinforcement learning is suitable for training this kind of neural network (Vantonen et al., 1990). The authors think that the evolutionary learning or genetic algorithm is a potential candidate. We are turning our attention to this aspect. Finally, it should be noted that molecular electronics may provide a simple and effective approach to implementing the neuron model with dynamically adjustable threshold and refractory periods, because there exist some artificial
biological or organic thin films which demonstrate characteristics similar to those of the neuron model proposed in this paper (Drain et al., 1989; Michaile et ai., 1990). References Drain, C.M. et al., 1989, Photogating of ionic currents across the lipid bilayer. Proc. IEEE EMBS l l t h Annual Int. Conf. 1336-1337. FitzHngh, R., 1961, Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445 - 466. Gan, Q., Wei, Y. and Conrad, M., 1990, Locally connected multilayer neural networks consisting of enzymatic neurons. J. Circuits Systems 2(2), 223-228. Hodgkin, A.L. and Huxley, A.F., 1952, A quantitative description of membrane current and its applications to conduction and excitation in nerve. J. Physiol. 117, 500- 544. Horn, D. and Usher, M., 1989, Neural networks with dynamical thresholds. Phys. Rev. A. 40, 1036-1044. MacGregor, R.J. and Lewis, E.R., 1976, Neural Modeling (Plenum Press, New York and London). McCulloch, W.S. and Pitts, W., 1943, A logical calculus of the ideas imminent in nervous activity. Bull. Math. Biophys. 5, 115-133. Meatier, J.L. et al., 1991, Programmable impulse neural circuits. IEEE Trans. Neural Networks 2, 101-108. Michaile, S. et al., 1990, Chloride ion modulation of the f a s t photoelectric signal in halorhodopsin thin films. Proc. IEEE EMBS 12th Annual Int. Conf. 1721-1723. Murray, A.F. et al., 1991, Pulse-stream VLSI neural networks mixing analog and digital techniques. IEEE Trans. Neural Networks 2, 193-204. Sanchez-Sinencio, E., 1991, Guest editorial -- neural network circuit implementations. IEEE Trans. Neural Networks 2, 192. Vantonen, K. et al., 1990, Pulse-strcam neural networks and reinforcement learning. Proc. IJCNN'90, San Diego 2, 267 - 272. Yanai, H. et al., 1990, Associative memory network composed of neurons with hysteretic property. Neural Networks 3, 223-228.
137
Elsevier Scientific Publishers Ireland Ltd.
Neural modeling with dynamically adjustable threshold and refractory period Qiang Gan and YU Wei Department of BiomedicaJ:Engineering, Southeast University, Nanfing ~10018 (P.R. of China) (Received December 12th,~ 1991) (Revision received August llth, 1992)
A variant of the FitzHugh-Nagumo model is proposed in order to fully make use of the computational properties of intraneuronal dynamics. The mechanisms of threshold and refractory periods resulting from the double dynamical processes are qualitatively studied through computer simulation. The results show that the variant neuron model has the property that its threshold, refractory period and response amplitude are dynamically adjustable. This paper has also discussed some problems relating to collective property, learning and implementation of the neural network based on the neuron model proposed. It is noted that the implicit way to describe threshold and refractory period is advantageous to adaptive learning in neural networks and that molecular electronics probably provides an effective approach to implementing the above neuron model.
Keywo~ds: Neuron model; Refractory period; Threshold dynamics; Neural networks.
1. Introduction
Mathematical modeling for neural systems aims at understanding the mechanisms of biological neural systems and providing the basis for artificial neural network modeling. The first mathematical neuron model was proposed and applied to construct neural nets based on simple logic calculus in 1943 (McCulloch and Pitts, 1943), which have played a very important role in promoting studies in information science, automation and computer systems' theory and is often called the M-P model. In 1952, Hodgkin and Huxley put for~yard a set of equations to describe the excitation and conduction of neural impulses in the squid axon (Hodgkin and Huxley, 1952), which is still the most widely accepted standard mathematical model for biological neurons. The Hodgkin-Huxley equaCarrespondenee to: Qiang Gan, Department of Biomedical Engineering, Southeast University, Nanjing 210018, P.R. of China.
tions (H-H equations) constitute a very complicated four-dimensional dynamical system. Either from the viewpoint of applications or from that of mathematical analysis, it is quite necessary to simplify the equations while mainraining the basic computational properties. Since then there has been a lot of research on the simplification and improvement of the H-H equations (MacGregor and Lewis, 1976), among them the FitzHugh-Nagumo model (FitzHugh, 1961), with the simplest mathematical expression, is the most widely acceptable simplified model of the H-H equations and is highly recommended for its potential applications in artificial neural network modeling (Sanchez-Sinencio, 1991). In recent years, scientists have turned much of their attention to the architecture of interconnection between neurons, but a little to the neuronal modeling itself. However, more and more evidence shows that intraneuronal dynamics contribute a great deal to collective properties of neural nets (e.g. Horn and Usher, 1989; Yanai, 1990).
0303-2647/92/$05.00 © 1992 Elsevier ScientificPublishers Ireland Ltd. Printed and Published in Ireland
138 The non-linear properties such as threshold and refractory periods play a key role in the information processing of biological neural systems. However, the M-P model has been used in most artificial neural networks, in which only the time-invariant threshold property is considered. It is shown that introducing a timevariant threshold function into the neural network model allows for the improvement of the information storage capacity and associative ability of the neural network (Horn and Usher, 1989; Yanal, 1990). In fact, the refractory period is equivalent to the time-variant threshold effect, which is advantageous to improving the performance of neural networks. The authors think that the refractory period expressed in an implicit (or distributed) way is more advantageous to the adaptive learning in neural nets and is also in accordance with the information processing mode in biological neural systems. In this respect, the FitzHugh-Nagumo model would be considered a good candidate (Sanchez-Sinencio, 1991), but the dynamical range of the refractory period in this model is very much limited. Especially, when the refractory period is shortened to some extent by adjusting certain parameters in the model, the threshold effect will no longer exist. On the other hand, in the practical application of neural networks different tasks often require different ranges of refractory periods (Horn and Usher, 1989; Yanai, 1990). Our paper presents a variant of the FitzHugh-Nagumo model, in which the dynamical range of threshold, refractory period and response amplitude can be arbitrarily adjusted and satisfy practical requirements.
2. Qualitative analysis and mathematical description So far the H-H equations are the most satisfactory mathematical model for biological neurons. The FitzHugh-Nagumo model, a two~limensional dynamical system, is the simplest continuous mathematical model, basically preserving the computational properties of the H-H equations. The model is described by the
following equations:d y / d t ffi - ( x - a + by)/c
(1)
d~ldt ffi c(y + x - X3/3 + Z)
(2)
where x and y are dynamical variables, z represents the stimulus and a, b and c are control parameters. Our computer simulation analysis shows that in order to produce threshold and refractory effect, the variables x and y should interact with each other and their change rates have to be considerably different. In addition, if one variable represents the excitation in response to the stimulus, the other has to represent the inhibition (or accommodation). The FitzHughNagumo model satisfies the above conditions, but remains somewhat deficient. Although the refractory period can be adjusted by varying the parameter c in the model, its dynamical range is very much limited. Because c controls the speeds of two dynamical processes, when c is near to or less than one, the speed difference of the two dynamical processes is not obvious. As a result, the threshold effect disappears in the FitzHugh-Nagumo model. This is shown in Fig. 1, which includes the x , y trajectory and two lines (x-nullcline and y-nullcline) representing d x / d t = 0 a n d d y / d t = O, respectively. In Fig. l(a) the stimulus is too weak to cross the xnullcline. In Fig. l(b) the stimulus is strong enough to cross the x-nullcline. If a threshold effect exists, the response should appear to be a sudden increase when the stimulus is strong enough to cross the x-nullcline. However, because c equals 1 in Fig. 1, there is no such sudden increase in the response to the stimulus, no matter whether the stimulus is weak or strong. The computer simulation shows that the shortest refractory period (including relative refractory period) of the FitzHugh-Nagumo model is around 5 ms.. As a matter of fact, one of a general biological neuron is 1 or 2 ms. and in the artificial neural network study a large dynamical range of refractory period could be expected (Horn and Usher, 1989). In addition, the response amplitude in the FitzHugh-
139
Parameters: a=-I b=2 c=1
Y 1
/
d y / d t = O: / = (a - x)/b
f
d x / d t = O: y=x,x,x/3-x
-1
Please input pulse amplitude: Z = 1 Please input the interval between pulses (ms): invl = 20 Please input the number of pulses: count = 1 Press space key to exit I Press Y to change constants 1 Press any other keys to continue I
Parameters: a=-I b=2 c : 1
-'~
d y / d t = O: y = (a - x)/b dx/dt = O: y=x,x,x/3-x
"
ly
(//i
b
x
-1L
Please input pulse amplitude: Z = 1.5 Please input the interval between pulses (ms): invl = 20 Please input the number of pulses: count = 1 Press space key to exit I Press Y to change constants I Press any other keys to continue I
Fig. 1. Effect of parameter c in Fizhugh-Nagumo model on threshold. (a) S t h n u h s is too weak to cross x-nullcline; (b) stimulus is strong enough to cross x-nullcline. In both cases there is no sudden change in response after stimulation because c ffi 1.
Nagumo model mainly depends on the shape of the x-nullcline, which is fixed so that the response amplitude iis basically unadjustable. Based on the above analysis, we propose the following differential equations as a variant in order to overcome the deficiency of the FitzHugh-Nagumo nlodel: dy/dt ffi -0.5 M (At
+
A2x + Y)
(3)
d~/dt = 2 M(x
-
Bx 3
+
Cy) + Ds
(4)
where x and y represent excited and inhibited variables respectively, s represents the stimulus and A1, A2, B, C, D, M are control parameters. In the FitzHugh-Nagumo model, the threshold effect is emphasized, while the refractory period is somewhat neglected. In our variant model, the refractoriness and its linear adjustability are
140 two of the most important properties of the neuron model. The variation of the variables x and y depends mainly on the shape and position of the nullclines which are described by the following equations: y =
-
(A 1 + A2X )
(5)
- Bx3)/C
(6)
y = - (x
The crossing point of the nullclines is the equilibrium point of the system. Obviously, parameter -A1 is the intercept of the y-nullcline at the y axis, A2 controls its slope. From Eqns. 3 and 4 we also know that the sign reversal of A1 changes the sign of x, y and s to the reverse. Parameters B and C affect the shape of the xnullcline so that the amplitudes of x and y can be controlled, where C only controls the dilation of the x-nullcline in the y direction and B mainly controls the dilation in the x direction. It is easy to find out from Eqns. 3 and 4 that parameter M controls the change rates of x and y, thus controlling the duration of the refractoriness and no matter what the value of M is, the difference between the change rates of x and y is large enough to give rise to the threshold effect. In addition, the dynamical range of the threshold can be adjusted by parameter D. In the following section, more detailed results are given through computer simulation analysis. 3. Computer simulation analysis Because of the non-linearity in the model, the variation rule of x and y is very complicated and it is quite difficult to get the precise solution to Eqns. 3 and 4. Based on computer simulation, we can analyze the stimulus-response relationship, represented by the carves of x and y against time in response to the stimuli or the x , y trajectory in the x-y plane. Therefore, we can analyze the threshold and refractory effect of the two-dimensional dynamical system described by Eqns. 3 and 4. Especially, the influence of parameters in Eqns. 3 and 4 on threshold and refractory period can be analyzed in this way.
In the study of the curves of x and y against time, emphasis is placed on the responses to two adjacent impulse stimuli. The change rates of x and y, the duration of refractoriness and the influence of the parameters can be observed clearly from these curves. In the x - y plane, we mainly pay attention to the effect of the parameters on the shape and position of the nullclines. Therefore, it is easy to observe the equilibrium point and threshold effect in the x - y plane. With a set of typical parameter values, the curves of x and y against time and the trajectory in the x-y plane are shown in Figs. 2(a) and (b), respectively. In Fig. 2(a) the curve of x is plotted in a slender line and y in a thick line. In Fig. 2(b) the nullclines are plotted in slender line and the z,y trajectory in thick line. In the following figures, the convention about the slender line and thick line is the same as that in Fig. 2. In addition, the parameter values, stimulus strength and interval between two adjacent impulse stimuli are also given in Fig. 2. The typical parameter values are as follows: A1 = 0.5, A2-- 1, B = 1, C-- 0.5, D = 1, a n d M = 1. As shown in the following figures, some parameter values are changed and the results are compared with those from the typical parameter values so that the effect of these parameters on the performance of the model can be analyzed. Figure 3 demonstrates the effect of parameters A1 and A2 on the equilibrium point. When the values of A1 and A2 are changed from their typical values to 0.25 and 0.5, respectively, the position of the y-nullcline changes considerably. As a result, three equilibrium points exist in the system. When the system is stimulated at a certain equilibrium point, it probably becomes stable again at another equilibrium point. As shown in Fig. 4, parameter B mainly affects the shape of the x-nullcline, thus the amplitudes of x and y and the duration of refractoriness, when B increases, the shape of the xnullcline is compressed in both x and y directions, otherwise it is stretched. However, the dilation in the x direction is much more obvious than that in the y direction. In addition, by
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Do you test pulse stimulation (y/n)? y Parameters: A 1 = 0.5 A 2 = 1 B= 1 C=0.5 D=I M=I
't x
x__
Z
W
1 0
dx/dt = 2M (x- B * x , x,x+C,y+D,S)
-1 dy/dt =
- 0 . 5 M (A1 x + y)
+
A2 * 0
8
16
t(ms)
|
I
24
32
.:1
2
Please input pulse amplitude: s = 0.8 Please input the time of next pulses (ms): T = 12 Press space key to exit I Press Y to change constants I Press any other keys to continue !
Parameters:
Y
A 1 = 0.5
A 2 -- 1
B= 1 D=I
C=0.5 M=I
dx/dt = 2M(x B,x,x,x+C,y +D,s)
t
-2
;
X
/"
dy/dt =
-0.SM(A 1 + A 2 * x+
y)
Please input pulse amplitude: s = 0.8 Please input the interval between pulses (ms): invl = 20 Please input the number of pulses: count = 1 Press space key to exit ! Press Y to change constants t Press any other keys to continue I
Fig. 2. Sfimulus-respon~,;e relationship of the improved model with a set of typical parameter values. (a) Curves of z and y against time; (b) trajectory in z ~ plane.
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Parameters: A 1 = 0.25 A 2 = 0.5 B= 1 C=0.5 D=I M=I
Y
iI !I
ii
dx/dt = 2 M ( x B,x,x,x+C,y+ D,s)
i
X
2
dy/dt = - 0 . 5 M (,41 + A 2 , x+y)
Please input pulse amplitude: s = - 0 . 5 Please input the interval between pulses(ms): invI = 20 Please input the number of pulses: count = 1 Press space key to exit I Press Y to change constants i Press any other keys to countine I
Fig. 3. Effect of parameters A 1 and A 2 on equilibrium point and threshold.
Param~ers: A1=0.5 A2=1 B=3 C=0.5 D=I M=I
Y
,I
dx/dt = 2M(x B,x,x,x+C, +D,s)
dy/dt= - 0 5M(A1 + A2 =
y
I
-2
X
2
/
,L\
x+y)
Please input pulse amplitude: s = 0.8 Please input the interval between pulses (ms): invl = 20 Please input the number of pulses: count = 1 Press space key to exit I Press Y to change constants ! Press any other keys to countine !
Fig. 4. Effect of parameter B on shape of z-nullcline. When B increases, x-nullcline is compressed so that response amplitude descends.
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carefully analyzing Eqns. 3 and 4, we can discover that when B is too small the shape of xnullcline is dominated by the linear term in Eqn. 6 so that the model can no longer simulate the properties of neurons. The effect of parameter C is very obvious, observed directly from Eqn. 6. When C increases, the x-nullcline is compressed only in the y direction, otherwise it is stretched. Therefore, the amplitudes of the x-nullcline in the x and y axes can be arbitrarily adjusted by combining the variation of B and C. The effect of parameter M can be clearly observed from the curves of x and y against time, as shown in Fig. 5. The parameter values are also listed in the figure. Parameter M only affects the change rates of x and y and thus the duration of the refractoriness but exerts no influence on the amplitudes of x and y and the threshold. The duration of refractoriness can be linearly adjusted by varying parameter M. When M increases, the refractory period is linearly shortened, otherwise it is extended.
The above analysis shows that the parameters in Eqns. 3 and 4 have obvious impact on the performance of the model. The parameter values can be adjusted to make the model satisfy any given targets of performances such as threshold and refractoriness, but it is impossible to do so in the FitzHugh-Nagumo model. For example, parameters B and C are firstly adjusted to guarantee the amplitudes of x and y. Next, A1 and A2 are adjusted to select a suitable equilibrium point. Then, D is adjusted to select any given threshold value. Finally, M can be adjusted to select a required refractory period. 4. Discussion and conclusion
Based on the above analysis, we think that the essence of the refractory period in neurons lies in the interaction of double (or multiple) dynamical processes with different speeds, which is in accordance with the viewpoint of neurobiology. Therefore, in order to obtain the refractory property in artificial neurons it is
Do you test pulse stimulus (y/n)? y
Parameters: : A 1 = 0.5
A2 = 1
B=I
C=0.5
D=I
M=4
x
_y
1'6
/(ms)
dx/dt = 2M(xB,x,x,x+C,y +D,s) dy/dt = - 0 . 5 M (A1 + A2 * x + y)
-2
1
0
8
24
312
Please input pulse amplitude: s = 0.8 Please input the time of next pulses (ms): T = 3 Press space key to exit ! Press Y to change constants I Press any other keys to continue I
Fig. 5. Effect of parameter M on refractory period. Refractory period can be arbitrarily adjusted by varying M. When M increases, duration of refractoriness is linearly shortened, otherwise it is extended.
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necessary to introduce internal dynamics into neuron models, that is, there should be at least one internal variable to connect the stimulus to the response. A neuron model with dynamically adjustable threshold, refractory period and response amplitude is proposed in this paper, which could be used in artificial neural networks. It is shown that the refractory period (or time-variant threshold effect) plays a very important role in improving the collective properties of neural networks (Gan et al., 1990; Horn and Usher, 1989; Yanai, 1990). However, there still is a lack of sound theoretical analysis and wide experimental findings, thus leaving much to be desired in promoting the application of refractory property in artificial neural networks. Because the refractory property is essentially characteristic of the time domain, it is profitable to make use of the refractory period in pulseencoding neural network architecture. In addition, this kind of architecture has the features of saving power and space and combining the merits of digital and analogue signal processing (Meader et al., 1991; Murray et al., 1991). So far there are still some problems to be solved for learning in the neural network of this architecture. It was noted that the reinforcement learning is suitable for training this kind of neural network (Vantonen et al., 1990). The authors think that the evolutionary learning or genetic algorithm is a potential candidate. We are turning our attention to this aspect. Finally, it should be noted that molecular electronics may provide a simple and effective approach to implementing the neuron model with dynamically adjustable threshold and refractory periods, because there exist some artificial
biological or organic thin films which demonstrate characteristics similar to those of the neuron model proposed in this paper (Drain et al., 1989; Michaile et ai., 1990). References Drain, C.M. et al., 1989, Photogating of ionic currents across the lipid bilayer. Proc. IEEE EMBS l l t h Annual Int. Conf. 1336-1337. FitzHngh, R., 1961, Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1, 445 - 466. Gan, Q., Wei, Y. and Conrad, M., 1990, Locally connected multilayer neural networks consisting of enzymatic neurons. J. Circuits Systems 2(2), 223-228. Hodgkin, A.L. and Huxley, A.F., 1952, A quantitative description of membrane current and its applications to conduction and excitation in nerve. J. Physiol. 117, 500- 544. Horn, D. and Usher, M., 1989, Neural networks with dynamical thresholds. Phys. Rev. A. 40, 1036-1044. MacGregor, R.J. and Lewis, E.R., 1976, Neural Modeling (Plenum Press, New York and London). McCulloch, W.S. and Pitts, W., 1943, A logical calculus of the ideas imminent in nervous activity. Bull. Math. Biophys. 5, 115-133. Meatier, J.L. et al., 1991, Programmable impulse neural circuits. IEEE Trans. Neural Networks 2, 101-108. Michaile, S. et al., 1990, Chloride ion modulation of the f a s t photoelectric signal in halorhodopsin thin films. Proc. IEEE EMBS 12th Annual Int. Conf. 1721-1723. Murray, A.F. et al., 1991, Pulse-stream VLSI neural networks mixing analog and digital techniques. IEEE Trans. Neural Networks 2, 193-204. Sanchez-Sinencio, E., 1991, Guest editorial -- neural network circuit implementations. IEEE Trans. Neural Networks 2, 192. Vantonen, K. et al., 1990, Pulse-strcam neural networks and reinforcement learning. Proc. IJCNN'90, San Diego 2, 267 - 272. Yanai, H. et al., 1990, Associative memory network composed of neurons with hysteretic property. Neural Networks 3, 223-228.