Nonlinear dynamical system approaches towards neural prosthesisHiroyuki Torikai and Sho Hashimoto Citation: AIP Conf. Proc. 1339, 78 (2011); doi: 10.1063/1.3574846 View online: http://dx.doi.org/10.1063/1.3574846 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=1339&Issue=1 Published by the American Institute of Physics. Additional information on AIP Conf. Proc.Journal Homepage: http://proceedings.aip.org/ Journal Information: http://proceedings.aip.org/about/about_the_proceedings Top downloads: http://proceedings.aip.org/dbt/most_downloaded.jsp?KEY=APCPCS Information for Authors: http://proceedings.aip.org/authors/information_for_authors

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Nonlinear dynamical system approaches towardsneural prosthesis

Hiroyuki Torikai and Sho Hashimoto

Graduate School of Engineering Science, Osaka University, Japan

Abstract. An asynchronous discrete-state spiking neurons is a wired system of shift registers thatcan mimic nonlinear dynamics of an ODE-based neuron model. The control parameter of the neuronis the wiring pattern among the registers and thus they are suitable for on-chip learning. In this paperan asynchronous discrete-state spiking neuron is introduced and its typical nonlinear phenomena aredemonstrated. Also, a learning algorithm for a set of neurons is presented and it is demonstrated thatthe algorithm enables the set of neurons to reconstruct nonlinear dynamics of another set of neuronswith unknown parameter values. The learning function is validated by FPGA experiments.

Keywords: Neuron ModelPACS: 05.45.-a

INTRODUCTION

The neural prosthesis means substitution of damaged biological neurons by artificialelectronic devices. Neural prostheses for some sensory neural systems have been al-ready in practical use, e.g., cochlea implant [1][2]. In addition, recently, fundamentalstudies on neural prostheses for central nerve systems have begun, e.g., VLSI artificialhippocampus slice and its implantation into living mouse [3][4]. A typical approach ofthe conventional neural prosthesis is to mimic responses of biological neurons based ontheir input-output relations of firing rates. That is, the conventional approach ignorestemporal spike patterns generated by nonlinear dynamics of neurons that are believed toplay important roles in the information processing of biological neural systems [5][7]. Asensory neuron converts analog signals (e.g., sound and light) into spike-trains, and thusit can be regarded as a kind of analog-to-spike (i.e., analog signal to digital bit stream)dynamic converter. From an electronic circuit viewpoints, a sensory neuron should bemodeled by an analog circuit which can accept analog inputs, e.g., our group has pre-sented a chaotic nonlinear electronic circuit model of spiral ganglion cell in the mam-malian inner ear [8]. A cortical/hippocampal neuron accepts spike-trains from the sen-sory and other cortical/ hippocampal neurons and generates a spike-train, and thus itcan be regarded as a kind of spikes-to-spike (i.e., multiple digital bit streams to singledigital bit stream) dynamic converter. Hence, it may be reasonable to model a corti-cal/hippocampal neuron not only by a pure analog circuit but also by an analog-digitalhybrid electronic circuit. Our group has been presented analog-digital hybrid electroniccircuit models of cortical/hippocampal neurons [9]-[13]. Since some of these modelscan be implemented by asynchronous digital circuits (where the asynchronous proper-ties correspond to analog dynamics), they are referred to as asynchronous discrete-statespiking neurons (ab. ADSNs). The ADSN is conceptually defined as follows:

International Conference on Applications in Nonlinear Dynamics - ICAND 2010AIP Conf. Proc. 1339, 78-87 (2011); doi: 10.1063/1.3574846

© 2011 American Institute of Physics 978-0-7354-0894-4/$30.00

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(1) the ADSN is a wired system of shift registers and thus it can be regarded as a specialkind of asynchronous cellular automaton (ab. CA);(2) the ADSN is designed to exhibit neuron-like behaviors (e.g., integrate-and-fire andresonate-and-fire behaviors) of ODE-based neuron models; and(3) the ADSN is designed to mimic various neuron-like bifurcations and responses ofODE-based neuron models.

In this paper, one of the ADSNs [11] is introduces and some recent new results arepresented as the followings. First, the ADSN is introduced and its neuron-like behaviorsare explained. Second, a fundamental problem toward neural prosthesis is stated: ap-proximation of nonlinear dynamics of a network of biological neurons by a network ofADSNs. In order to approach such a general problem, in this paper, a hardware-orientedlearning algorithm is proposed so that a set of ADSNs (students) can reconstruct un-known nonlinear dynamics of another set of ADSNs (teachers). It is shown that thealgorithm enables the students to reconstruct the unknown nonlinear dynamics of theteachers. Third, the learning function is validated by FPGA experiments. The signifi-cances of the ADSN include the following points.• The ADSN is a wired system of shift registers and its learning algorithm is based on theupdates of the wiring pattern. Hence, the ADSN and its learning algorithm are suitablefor implementation on a reconfigurable digital circuit such as a dynamic FPGA. Froma neural prosthesis viewpoint, this means that the response characteristics of the ADSNcan be re-adjusted after implantation. From an artificial pulse-coupled neural networkviewpoint, this means that our approach will provide feasible solutions for developing apulse-coupled neural network with on-chip learning capabilities. On the other hand, it istroublesome to implement a dynamic parameter update rule of an analog spiking neuronmodel in an analog VLSI.• An FPGA-implemented ADSN is expected to be low energy consuming since at mostonly four digital memory cells (i.e., flip-flops) change their states at every time step.On the contrary, an FPGA-implemented numerical-integration-based neuron model isexpected to be higher energy consuming since a lot of memory cells change their statesat every time step. Also, since the control parameter of the ADSN is the wiring pattern,its learning can be implemented on an FPGA by using a simple discrete optimizationalgorithm [11] which is expected to be lower energy consuming than a floating-point-calculation-based learning algorithm such as the steepest gradient method. Such a lowenergy consuming ADSN with the on-FPGA learning capability will be applicable todevelop a future neural prosthesis chip with after-implant learning capability and todevelop a large-scale artificial pulse-coupled neural network with on-chip learning ca-pability.• The CA is an important class of nonlinear dynamical system [14]. From an academicviewpoint, it is important and interesting to develop spiking neuron models not only by anonlinear ODE as is usually done but also by a CA. Note that meaningful example sys-tems have played crucially important roles in the history of development of nonlineardynamical system theory [15]-[17]. The RDN may be one of such meaningful examplesystems for developing the theory of CA.

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1p

p

0p 0x0r

Y

r

(a)

0l

Reconfigurablewires

p-cells x-cells

1−M

p

1−Nlx

1−N

i jx

1−M

M N

0r

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1x

)(tSInput Spike-train

0b

b 1−N

Internal Clock )(tCL

1p

p

0p 0x0r

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r

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p-cells x-cells

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p

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0r

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1x

)(tSInput Spike-train

0b

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t

)(tS

0

(b)

d

tM2

01

M3

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M0

)(: tB

)(: tX

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01

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t

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)1( +nt)(nt

)(n∆

)(tC

1 t

)(tS

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)(: tB

)(: tX

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01

)(tX

M3

1−N

(c)

M0

t

Y

0

(d)

)1( +nt)(nt

)(n∆

)(tC

1

FIGURE 1. (a) An example of the ADSN. (b) Input spike-train S(t) with period d. (c) Internalclock CL(t) with period 1 and basic dynamics of the ADSN. (d) Output spike-train Y (t).

ASYNCHRONOUS DISCRETE-STATE SPIKING NEURON

Fig.1(a) shows an example of the asynchronous discrete-state spiking neuron (ab.ADSN). The ADSN consists of three parts: M pieces of p-cells that are indexed byi∈ {0,1, · · · ,M−1}, M≥ 2; N pieces of x-cells that are indexed by j ∈ {0,1, · · · ,N−1},N ≥ 2; and reconfigurable wires from the p-cells to the x-cells. First, we explain the dy-namics of the p-cells. Each p-cell has a binary state pi(t) ∈ {0,1}, where t ∈ [0,∞)represents continuous time. In this paper, we assume that one p-cell has a state pi(t) = 1and the other p-cells have states pk(t) = 0, k 6= i. Under this assumption, we can repre-sent the state vector (p0(t), · · · , pM−1(t)) by the following scalar integer P(t).

If pi(t) = 1 then P(t) = i, P(t) ∈ {0,1, · · · ,M−1}.

We can assume P(0) = 0 without loss of generality. We refer to P(t) as a state of thep-cells. The p-cells commonly accept the following internal clock CL(t):

CL(t) ={

1 if t = 0,1,2, · · · ,0 otherwise,

which has the normalized period of 1. The dynamics of the p-cells is then described by

P(t +1) = P(t)+1 (mod M). (1)

The state P(t) oscillates periodically with period M. Second, we explain the reconfig-urable wires. The binary state vector (p0(t), · · · , pM−1(t)) ∈ {0,1}M of the p-cells isinput to the left terminals (l0, · · · , lM−1) of the wires. Each left terminal li has one wireto one of the right terminals (r0, · · · ,rN−1), and each right terminal r j can accept an ar-bitrary number of wires. In order to describe the patterns of the wiring, we introduce thefollowing wiring function A : {0,1, · · · ,M−1}→ {0,1, · · · ,N−1}.

A(i)≡ j if li is wired to r j

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where “≡" is used to represent the “definition" hereafter. We represent the wiringfunction A(i) by the following parameter vector:

AAA≡ (A(0),A(1), · · · ,A(M−1)).

We refer to the parameter vector AAA as the wiring pattern. In the case of Fig.1(a), thewiring pattern is AAA = (0,1,2,3,3,2,1). As shown in this figure, the right terminals(r0, · · · ,rN−1) output a binary signal vector (b0(t) , · · · , bN−1(t)) ∈ {0,1}N . Sinceone element b j(t) is 1 and the other elements bk(t), k 6= j, are 0, the signal vector(b0(t), · · · ,bN−1(t)) can be represented by the following scalar integer B(t).

If b j(t) = 1 then B(t) = j, B(t) ∈ {0,1, · · · ,N−1}.We refer to B(t) as a base signal. Using the wiring function A(i), the base signal B(t) isgiven by

B(t) = A(P(t)). (2)

In Fig.1(c) a base signal B(t) is illustrated by white circles. Third, we explain thedynamics of the x-cells. Each x-cell has a binary state x j(t) ∈ {0,1}. In this paper, weassume that one x-cell has a state x j(t) = 1 and the other x-cells have states xk(t) = 0,k 6= j. Under this assumption, we can represent the state vector (x0(t), · · · ,xN−1(t)) bythe following scalar integer X(t):

If x j(t) = 1 then X(t) = j, X(t) ∈ {0,1, · · · ,N−1}.In this paper we assume that xN−1(0) = 1 and x j(0) = 0 for all j 6= N−1 (i.e., X(0) =N − 1). We refer to X(t) as a state of the x-cells. From a neuron model viewpoint,the state X(t) is regarded as a membrane potential of an integrate-and-fire neuronmodel [7]. Moreover, as shown in Fig.2(a), the x-cells accept the binary signal vector(b0(t), · · · ,bN−1(t)) ∈ {0,1}N . In addition, the x-cells commonly accept an input spike-train S(t). In this paper, we focus on a periodic input

S(t) ={

1 if t = 0,d,2d, · · · ,0 otherwise,

with a period d as shown in Fig.1(b). From the neuron model viewpoint, the input S(t)is regarded as a stimulation current [7]. Now the dynamics of the x-cells is described by

X(t +d) ={

X(t)+1 if X(t) 6= N−1, (shift)B(t +d) if X(t) = N−1. (reset) (3)

In Fig.1(c), a typical waveform of the state X(t) is illustrated by black boxes. If theblack box (which corresponds to a membrane potential) reaches the top position (whichcorresponds to a firing threshold) at t = t(n), the black box at t = t(n) + d is reset tothe position of the white circle at t = t(n)+ d. At t = t(n), the ADSN outputs a spikeY (t(n)) = 1. After the reset, the black box is shifted upward. Repeating such shift-and-reset dynamics, the ADSN generates the following spike-train Y (t) as shown in Fig.1(d):

Y (t)≡{

1 if X(t) = N−1,0 if X(t) 6= N−1,

t = 0,d,2d, · · · . (4)

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0 2d

M(a) )(nθSpike phase

Input period

0 2dM∆

Average ISI(b)

0 2d

M(a) )(nθSpike phase

Input period

0 2dM∆

Average ISI(b)

FIGURE 2. (a) and (b) show a bifurcation diagram of the spike phase θ(n) and the characteris-tics of the average ISI ∆ of the ADSN, respectively. The ADSN is characterized by the parametervalues in Equation (5).

The shift-and-reset dynamics corresponds to the integrate-and-fire dynamics of theneuron model [7]. As a result, the dynamics of the ADSN is described by Equations(1), (2), (3) and (4), where the ADSN and input spike-train S(t) are characterized by theparameters (M,N,AAA,d).

We now show typical examples of the bifurcation phenomena and response charac-teristics of the ADSN. As shown in Fig.1(d), we denote the n-th spike position by t(n).We then introduce the following variable θ(n) on a co-ordinate θ ≡ t (mod M).

θ(n)≡ t(n) (mod M).

In this paper, we refer to θ(n) as the n-th spike phase with respect to the period M of thebase signal B(t). Fig.2(a) shows a bifurcation diagram of the spike phase θ(n) for theperiod d of the input spike-train S(t), where the ADSN is characterized by the followingparameter values.

M = N = 32, A(i) = Int(11.5−11sin(2πi/32)) (5)

where i = 0,1, · · · ,31, and the function Int(x) gives the integer part of a real number x.As shown in Fig.1(d), we denote the n-th inter-spike interval (ISI) by

∆(n)≡ t(n+1)− t(n).

We then define the following average ISI ∆:

∆≡ 1α

α

∑n=1

∆(n) (6)

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where α is an appropriately large number of ISIs for averaging. Fig.2(b) shows the char-acteristics of the average ISI ∆ for the input period d, where the ADSN is characterizedby Equation (5). We regard the characteristics of ∆ as the response characteristics of theADSN to the input spike-train S(t) since the firing rate (= ∆−1) is one of the most fun-damental characteristics for investigating the pulse coding abilities of a neuron model.We emphasize that the ADSN can exhibit various bifurcation phenomena and responsecharacteristics when the wiring pattern AAA is adjusted.

ON-FPGA LEARNING

Problem statement and Learning algorithm

In this section, we consider the following problem as a fundamental research towardthe neural prosthesis, where a tilde “˜" is used to represent a “teacher" hereafter.

Problem Statement: Can a set of ADSNs (students) reconstruct or approximate nonlin-ear dynamics and response characteristics of another set of ADSNs (teachers) with un-known wiring patterns (AAA1

, · · · , AAAK) by using the multiplexed spike-trains (U1, · · · ,UL)and a learning algorithm that utilizes successive updates of the students’ wiring patterns(AAA1, · · · ,AAAK)?

In this paper we consider the case of L = 1. Then the multiplexed spike-train is givenand described by

U(t) = Y 1(t)+ Y 2(t)+ · · ·+ Y K(t) (7)

where + denotes the OR operation. We refer to U(t) as the multiplexed teacher spike-train. Similarly, a multiplexed student spike-train is give by

U(t) = Y 1(t)+Y 2(t)+ · · ·+Y K(t). (8)

In order to present a learning algorithm, we define the following cost C that measures thedifference between the multiplexed teacher spike-train U(t) and the multiplexed studentspike-train U(t).

C =IJ

∑m=1

U(tm)⊕U(tm), tm is such that S(tm) = 1 (9)

where ⊕ denotes the exclusive OR operation, and I and J are appropriate positiveintegers. The cost C is 0 if and only if the multiplexed student spike-train U(t) is identicalwith the multiplexed teacher spike-train U(t). Now we propose the following simpleFPGA-friendly learning algorithm which is a generalized one of our previous algorithm[11].

Learning algorithmStep 1: Initialize a counter to c := 0 which is used to count the number of learningiterations, where “ := ” represents the substitution of the right term into the left term

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hereafter. Initialize the wiring patterns of the K students to AAAk,old := (0, · · · ,0), k =1,2, · · · ,K.Step 2: Change one element in the wiring patterns (AAA1,old, · · · ,AAAK,old) randomly, anddenote them by (AAA1,new, · · · ,AAAK,new).Step 3: Create J ISIs (d(1), d(2), · · · ,d(J)) of the input spike-train S(t) randomly, whered( j) ∈ (dmin,dmax). Let I be a positive integer. Generate a spike-train S(t) whose ISIs are

d(1), · · · ,d(1)︸ ︷︷ ︸

I

, d(2), · · · ,d(2)︸ ︷︷ ︸

I

, · · · , d(J), · · · ,d(J)︸ ︷︷ ︸

I

.

Input the spike-train S(t) to both the K teachers and the K students.Step 4: Calculate the costs Cold and Cnew for the original wiring patterns(AAA1,old, · · · ,AAAK,old) and the new wiring patterns (AAA1,new, · · · ,AAAK,new), respectively. IfCnew ≤Cold , then go to Step 6. If Cnew > Cold , then go to Step 5.Step 5: Generate a random integer r ∈ [0,rmax]. If r +Cth < Cold , then go to Step 6.This is called compulsory-update. If r +Cth ≥Cold , then go to Step 7.Step 6: Update the wiring functions of the student to AAAk,old := AAAk,new, for k = 1, · · · ,K.Go to Step 7.Step 7: Increment the counter c by one. Let cmax be a given maximum counter value. Ifc≤ cmax, then go to Step 2. If c = cmax, then terminate the algorithm.

Learning characteristics

Fig.3 shows learning characteristics for K = 2. As examples of the teachers, we usethe ADSNs characterized by

M1 = N1 = 8, AAA111 = (4,2,1,2,4,6,7,6),

M2 = N2 = 8, AAA222 = (2,3,4,5,2,3,4,5).(10)

Fig.3(a) shows response characteristics of the two teachers. It can be seen that theresponse characteristics of the two students in Figs.3(b), (c) and (d) approach to those ofthe two teachers in Fig.3(a) as the learning proceeds. In order to characterize the learningresult, we define the following success rate γ of the learning.

γ ≡ Number of the learning trials that lead to C = 0Number of all the learning trials

.

Figs.3(e) and (f) show characteristics of the cost C and the success rate γ for 1000learning trials, respectively. Fig.3(f) says that about 40% of the learning trials lead tothe zero cost C = 0 after the learning. Fig.3(e) says that, even if the cost C is not zero,typical learning trials lead to small cost C. The learning characteristics in Fig.3 suggestthat the learning algorithm enables multiple student ADSNs to reconstruct unknownnonlinear dynamics of multiple teacher ADSNs.

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0 21

16∆~Average ISI

dInput period

(a) Teachers’ responses160

0 21

16∆Average ISI

dInput period

(b) Students’ responses160

0 21

16∆Average ISI

dInput period

(d) Students’ responses160

0 21

16∆Average ISI

dInput period

(c) Students’ responses160

0 510

150Cost

cLearning iterationsC max.ave.min.

(e)

γ0 510cLearning iterations

1 (f)0 21

16∆~Average ISI

dInput period

(a) Teachers’ responses160

0 21

16∆Average ISI

dInput period

(b) Students’ responses160

0 21

16∆Average ISI

dInput period

(d) Students’ responses160

0 21

16∆Average ISI

dInput period

(c) Students’ responses160

0 510

150Cost

cLearning iterationsC max.ave.min.

(e)

γ0 510cLearning iterations

1 (f)

FIGURE 3. Learning characteristics for K = 2. The teachers are characterized by Equa-tion (10). The learning algorithm is characterized by (dmin,dmax,L,J,rmax,Cth,cmax) =(0,2,26,10,10,105). (a) shows the response characteristics of the teachers. (b) shows the re-sponse characteristics of the students just after the initialization (c = 0). The cost is C = 121. (c)shows the response characteristics of the students after c = 10000 learning iterations. The cost isC = 22. (d) shows the response characteristics of the student after the learning (c = cmax = 105).The cost is C = 0. (e) The characteristics of the cost C for the learning iteration c. The graphsshow the maximum, the average and the minimum for 1000 learning trials. (f) The characteristicsof the success rate γ .

On-FPGA learning

We have implemented a teacher, a student and essential parts of the learning algorithmon a FPGA. The teacher is characterized by

M1 = N1 = 8, AAA111 = (4,2,1,2,4,6,7,6). (11)

Fig.4 shows experimental measurements of the learning for K = 1. It can be confirmedthat the student spike-train Y is identical with the teacher spike-train Y for variousperiods d of the stimulation input S. These results show that the on-FPGA learningenables the on-FPGA student to reconstructs response characteristics of the teacher.

CONCLUSION

It was shown that the ADSN can exhibit various bifurcation phenomena and responsecharacteristics to the input spike-train. It was also shown that, using the hardware-oriented learning algorithm, the set of ADSNs can reconstruct the response characteris-tics of another set of ADSNs with unknown wiring patterns. The learning function wasvalidated by the FPGA experiments. Future problems for neural prosthesis are including

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[ms]2

Y~

Y

Y~

Y

Y~

Y

(a)

(b)

(c)

t[ms]2

Y~

Y

Y~

Y

Y~

Y

(a)

(b)

(c)

t

FIGURE 4. Experimental measurements for K = 1. A teacher characterized by Equation (11) andthe learning algorithm characterized by (dmin,dmax,L,J,rmax,Cth,cmax) = (0,2,26,10,10,105) are imple-mented in a Xilinx’s FPGA XC2S100. (a), (b) and (c) show the output spike-trains Y (t) of the teacherand Y (t) of the student after learning, where a positive edge corresponds to a spike. The input period d isapproximately (a) 0.62[msec], (b) 0.83[msec] and (c) 1.25[msec].

the following points. (a) Development of the ADSN which can reconstruct more vari-ous neural behaviors of biological neurons. In [12], our group presents an ADSN thatcan reconstruct almost all excitatory responses of biological neurons. (b) Developmentof a mathematical analysis tool for the ADSN. In [11][12], our group present discrete-continuous hybrid return maps that can analytically describe the dynamics of ADSNs.Using the maps, nonlinear phenomena of the ADSNs can be analyzed. (c) Developmentof a pulse-coupled network of ADSNs and its learning algorithm so that the network canmimic nonlinear dynamics of a network of biological neurons. The learning algorithmproposed in this paper will be a starting point to approach such a big problem. The au-thors would like to thank Professor Toshimitsu Ushio of Osaka University for valuablediscussions.

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