6
ISSN 10642269, Journal of Communications Technology and Electronics, 2010, Vol. 55, No. 4, pp. 429–434. © Pleiades Publishing, Inc., 2010. Original Russian Text © A.S. Dmitriev, A.I. Ryzhov, 2010, published in Radiotekhnika i Electronika, 2010, Vol. 55, No. 4, pp. 459–464. 429 INTRODUCTION The question about organization of information processes in neuron systems is fundamental and draws attention of researchers for a long time. In this field, the first successful study was performed by A.L. Hodgkin and A.F Huxley [1] who have proposed a neuron model for central rhythm generators that control motions of liv ing organisms. Subsequently, other types of mathemati cal models, which describe neuron behavior [2–4], including spike generation and the neuron response to their action, have been developed. These simulation efforts have made it possible to understand the funda mental principles underlying the singleneuron dynamics and, partly, neuron nets. However, the question about data transmission implemented at the information level of neuron sys tems, rather than at the signal level, still remains unex plained. In this study, the problem of information transmission between neurons is investigated so as to reveal the possibility of organizing transmission of information streams in the form of bit streams. In con nection with this, the characteristics of the communi cations channel between neurons and their depen dences are of interest. 1. COMMUNICATIONS SYSTEM MODEL To solve the problem formulated above, the com munications system is simulated by a mathematical model of a transmitting neuron and the same model of a receiving neuron. In addition, the communications system model involves a former of the stream of rect angular pulses fed to the input of the neuron transmit ter. With the use of a certain encoding method, such a pulse stream is associated with the pulse stream trans mitted. In binary data transmission, modulation is performed so that “1” corresponds to the presence of a pulse in the certain time position, and “0,” to its absence during decoding. (In the literature, this mod ulation scheme is called onoff keying (OOK). The next communications element is the “channel” used for transmission of the output signal of the neuron transmitter. The spike stream generated by the trans mitting neuron passes through this channel and comes to the input of the neuron receiver. Its output is con nected to the decoding device. An information bit stream is assumed to be suc cessfully transmitted with the help of the “neuron communications channel,” if the decoding device correctly recovers the bit sequence coming to the input of the neuron transmitter in the form of a pulse sequence. Thus, it is of importance that a pulse sequence is recovered without errors and its recovery is barely dependent on the encoding method. However, note that the encoding method affects the noise immunity of a transmission system. The block diagram of the communications system under consideration is pre sented in Fig. 1. As a neuron model, we use the Hodgkin–Huxley model demonstrating the basic properties of a neuron from the standpoint of the response to external signals and spike generation. It is described by the combined equations (1) Here, C m is the membrane capacitance; V is the mem brane potential; I ion is the ion current through the membrane; dimensionless variables m, h, and n char acterize the membrane conductance; and coefficients C m V · I ion Vmhn , , , ( ) , = m · α m V ( ) 1 m ( ) β m V ( ) m, = h · α h V ( ) 1 h ( ) β h V ( ) h , = n · α n V ( ) 1 n ( ) β n V ( ) n . = Digital Data Transmission between Neuronlike Elements A. S. Dmitriev and A. I. Ryzhov Received May 4, 2009 Abstract—The problem of digital data transmission between two neuronlike systems has been investigated. The former of a sequence of rectangular pulses is used as an information signal source. The behavior of neu ronlike elements is described with the help of the Hodgkin–Huxley mathematical model. Such a system is shown to form the communications channel of fairly high quality in terms of the classical theory of informa tion transmission. DOI: 10.1134/S106422691004008X DYNAMICS CHAOS IN RADIOPHYSICS AND ELECTRONICS

Digital data transmission between neuronlike elements

Embed Size (px)

Citation preview

ISSN 1064�2269, Journal of Communications Technology and Electronics, 2010, Vol. 55, No. 4, pp. 429–434. © Pleiades Publishing, Inc., 2010.Original Russian Text © A.S. Dmitriev, A.I. Ryzhov, 2010, published in Radiotekhnika i Electronika, 2010, Vol. 55, No. 4, pp. 459–464.

429

INTRODUCTION

The question about organization of informationprocesses in neuron systems is fundamental and drawsattention of researchers for a long time. In this field, thefirst successful study was performed by A.L. Hodgkin andA.F Huxley [1] who have proposed a neuron model forcentral rhythm generators that control motions of liv�ing organisms. Subsequently, other types of mathemati�cal models, which describe neuron behavior [2–4],including spike generation and the neuron response totheir action, have been developed. These simulationefforts have made it possible to understand the funda�mental principles underlying the single�neurondynamics and, partly, neuron nets.

However, the question about data transmissionimplemented at the information level of neuron sys�tems, rather than at the signal level, still remains unex�plained. In this study, the problem of informationtransmission between neurons is investigated so as toreveal the possibility of organizing transmission ofinformation streams in the form of bit streams. In con�nection with this, the characteristics of the communi�cations channel between neurons and their depen�dences are of interest.

1. COMMUNICATIONS SYSTEM MODEL

To solve the problem formulated above, the com�munications system is simulated by a mathematicalmodel of a transmitting neuron and the same model ofa receiving neuron. In addition, the communicationssystem model involves a former of the stream of rect�angular pulses fed to the input of the neuron transmit�ter. With the use of a certain encoding method, such apulse stream is associated with the pulse stream trans�mitted. In binary data transmission, modulation isperformed so that “1” corresponds to the presence ofa pulse in the certain time position, and “0,” to itsabsence during decoding. (In the literature, this mod�

ulation scheme is called on�off keying (OOK). Thenext communications element is the “channel” usedfor transmission of the output signal of the neurontransmitter. The spike stream generated by the trans�mitting neuron passes through this channel and comesto the input of the neuron receiver. Its output is con�nected to the decoding device.

An information bit stream is assumed to be suc�cessfully transmitted with the help of the “neuroncommunications channel,” if the decoding devicecorrectly recovers the bit sequence coming to the inputof the neuron transmitter in the form of a pulsesequence.

Thus, it is of importance that a pulse sequence isrecovered without errors and its recovery is barelydependent on the encoding method. However, notethat the encoding method affects the noise immunityof a transmission system. The block diagram of thecommunications system under consideration is pre�sented in Fig. 1.

As a neuron model, we use the Hodgkin–Huxleymodel demonstrating the basic properties of a neuronfrom the standpoint of the response to external signalsand spike generation. It is described by the combinedequations

(1)

Here, Cm is the membrane capacitance; V is the mem�brane potential; Iion is the ion current through themembrane; dimensionless variables m, h, and n char�acterize the membrane conductance; and coefficients

CmV· Iion V m h n, , ,( ),=

m· αm V( ) 1 m–( ) βm V( )m,–=

h· αh V( ) 1 h–( ) βh V( )h,–=

n· αn V( ) 1 n–( ) βn V( )n.–=

Digital Data Transmission between Neuronlike ElementsA. S. Dmitriev and A. I. Ryzhov

Received May 4, 2009

Abstract—The problem of digital data transmission between two neuronlike systems has been investigated.The former of a sequence of rectangular pulses is used as an information signal source. The behavior of neu�ronlike elements is described with the help of the Hodgkin–Huxley mathematical model. Such a system isshown to form the communications channel of fairly high quality in terms of the classical theory of informa�tion transmission.

DOI: 10.1134/S106422691004008X

DYNAMICS CHAOS IN RADIOPHYSICS AND ELECTRONICS

430

JOURNAL OF CJOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 55 No. 4 2010

DMITRIEV, RYZHOV

αm, βm, αn, βn, αh, and βh determine the ion dynamics.The ion current through the membrane is defined as

(2)

Coefficients αm, βm, αn, βn, αh, and βh are calculatedfrom the empirical relationships

(3)

Iion V m h n, , ,( )

= GKn4 V VK–( )– GNam3h V VNa–( )– GL V VL–( ).–

αm 0.1 25 V–

25 V–10

������������⎝ ⎠⎛ ⎞exp 1–

��������������������������������, βm 4 V–18�����⎝ ⎠⎛ ⎞ ,exp= =

αh 0.07 V–20�����⎝ ⎠⎛ ⎞ , βhexp 1

30 V–10

������������⎝ ⎠⎛ ⎞exp 1+

��������������������������������,= =

αn 0.01 10 V–

10 V–10

������������⎝ ⎠⎛ ⎞exp 1–

��������������������������������, βn 0.125 V–80�����⎝ ⎠⎛ ⎞ .exp= =

Below, the following values of parameters are used incombined equations (1): Cm = 1 μF/cm2, the maximumNa+ conductance is GNa = 120 mS/cm2, the maximumK+ conductance is GK = 36 mS/cm2, the backgroundleakage conductance is GL = 0.3 mS/cm2, the equilib�rium potential for Na+ is VNa = 110 mV, the equilibriumpotential for K+ is VK = –12 mV, and the equilibriumpotential for the leakage channel is VL = 10.6 mV.

2. TRANSMITTING NEURON

In the presence of an external signal fed to the neu�ron receiver, its dynamics is described by combinedequations (1) with additive term Iext, which is intro�duced into the first equation to estimate the mem�brane current caused by an external action.

Figure 2 illustrates the neuron response to theexternal signal, a sequence of rectangular pulses. Theiramplitude is 4 μA/cm2. When the input signal ampli�

1 2

3 4 5

Pulse former

Neuron transmitter

Neuronreceiver

Noise

Thresholddevice

Fig. 1. Information transmission scheme with the use of neuronlike elements: (1) rectangular pulse stream, (2) spike sequencetransmitted, (3) signal propagation over the communications channel, (4) received spikes, (5) detected signal.

100

50

0

2001801601401201000 80604020–50

4

2

2001801601401201000 80604020

I, µA/cm2

V, mV

(а)

(b)

t, ms

t, ms

Fig. 2. Neuron response to the pulse action: (a) a sequence of rectangular pulses for I = 4 µA/cm2 and T = 20 ms and (b) theneuron membrane potential.

JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 55 No. 4 2010

DIGITAL DATA TRANSMISSION BETWEEN NEURONLIKE ELEMENTS 431

tude increases, the spike amplitude remains almostunchanged and reaches ~100 mV.

Since information is transmitted with the help ofspike generation, the characteristics of the communi�cations channel under consideration depend on theconditions that make it possible to implement suchgeneration in the presence of an external pulsed signal.

First of all, let us determine the amplitude andwidth of input pulses at which the neuron response is asequence of spikes. It is assumed that a spike isdetected in the system if an output pulsed signal hasthe amplitude exceeding V = 50 mV. For each of theinput pulse amplitudes, the pulse width is graduallyincreased and such an overshoot is registered. Simula�tion results have shown that the critical width corre�sponding to spike generation is Tcr = 2 ms if the inputsignal amplitude is I = 4 μA/cm2. When the inputpulse width is smaller, spikes are not observed at theneuron output. In the case of narrower rectangularpulses used to increase the transmission rate, outputspikes can be obtained only by increasing the inputpulse amplitude.

Figure 3 depicts the dependence between the max�imum amplitude Vmax of an output signal and theincoming pulse width T when the fixed amplitude ofan external action is I = 4 μA/cm2.

It is seen from this figure that, for the given value ofcurrent, the minimum width of the rectangular pulsefed to the input of a neuron transmitter, which enablesspike generation, is Tcr ~ 2 ms, while the spike width is~10 ms. Thus, it can be inferred that spikes at the out�put of the neuron receiver arise in response to spikescoming from the neuron transmitter and their widthdoes not interfere reception.

The neuron model respond to a sequence of inputrectangular pulses so that the corresponding sequenceof spikes appears at the neuron output only if the pulsespacing is sufficiently large. When the pulse spacing issmall, neurons themselves as well as the Hodgkin–Huxley model demonstrate the refractory effect [5]consisting in that a certain time period is necessary fora neuron to be excited after the previous externalaction. Thus, the refractory period determines theminimum time interval within which a neuron cannotgenerate a spike after previous generation. For theselected parameters of the model, refractory period Tr

is ~10 ms.

3. RECEIVING NEURON

Let us consider a neuron�like element operating asa receiver sensitive to spikes that are generated by aneuron transmitter and come to its input through thecommunications channel. It is assumed that an infor�mation signal source produces rectangular pulses withperiod T > Tr (Fig. 4a) and this pulse sequence is fed to

the input of a neuron transmitter. As a result, asequence of spikes (Fig. 4b) is generated at the outputof the neuron transmitter, propagates over the com�munications channel, and comes to the input of theneuron receiver. Thereafter, a sequence of spikesappears at the output of the neuron receiver (Fig. 4c).

Thus, in the absence of signal attenuation and dis�tortion, information (a sequence of spikes) is correctlytransmitted through the communications channel.However, in real communications channels, signalsare distorted by noise and their attenuation occurs inthe propagation medium. Let us consider the datatransmission system operating under the influence ofthese factors. Simulation of the neuron receiver hasrevealed that the output spike amplitude remains atapproximately the same level even if the input signal isattenuated several times. At the same time, the outputsignal decreases stepwise to 8 mV when the input sig�nal is attenuated ten times.

Since an information signal is transmitted underthe restrictions discussed above (first of all, in the pres�ence of a refractory period), correct signal reception isdetermined mainly by the amplitude of incomingspikes. For example, if the refractory periods of thetransmitter and receiver are identical, a signal is cor�rectly received by the corresponding neuron becausethe neuron transmitter cannot send two spikes that arevery closely spaced in time. Figure 5 illustrates the casewhere bit losses in the neuron transmitter are causedby small time intervals between rectangular pulses ofthe information source.

Let us investigate the influence of channel noise onoperation of the neuron receiver. Under the action ofan external noise signal, its response varies as follows.The small noise amplitude leads to the noisy outputsignal whose amplitude is also small. When the input

100

90

8070

60

50

40

30

20

6543210

10

Т, ms

Vmax, mV

Fig. 3. Dependence between the maximum amplitudeVmax of an output signal and the incoming pulse width T

for I = 4 µA/cm2 .

432

JOURNAL OF CJOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 55 No. 4 2010

DMITRIEV, RYZHOV

150

100

50

0

2001801601401201000 80604020–50

300

200

10

0

2001801601401201000 80604020–100

543

2

2001801601401201000 80604020

1

I, µA/cm2

V, mV t, ms

(а)

(b)

(c)

I, µA/cm2 t, ms

t, ms

Fig. 4. Neuronlike receiver response to incoming spikes: (a) rectangular pulses at the input of the neuronlike transmitter forT = 20 ms and I = 4 µA/cm2, (b) the input signal of the neuronlike receiver, and (c) the output signal of the neuronlike receiver.

150

100

50

0

2001801601401201000 80604020–50

300

200

10

0

2001801601401201000 80604020–100

543

2

200180160140120100806040200

1

V, mV

I, µA/cm2 t, ms

t, ms

t, ms

I, µA/cm2

(а)

(b)

(c)

Fig. 5. Neuronlike receiver response to incoming spikes: (a) rectangular pulses at the input of the neuronlike transmitter forT = 10 ms and I = 4 µA/cm2, (b) the input signal of the neuronlike receiver, and (c) the output signal of the neuronlike receiver.

JOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 55 No. 4 2010

DIGITAL DATA TRANSMISSION BETWEEN NEURONLIKE ELEMENTS 433

noise amplitude increases, several randomly locatedspikes appear in the system response, followed bymore frequent repetition of spikes. Figure 6 depicts theneuron response to Gaussian noise with the amplitudeon the order of 100 μA/cm2.

Let us consider the reception scheme in which auseful signal is distorted by noise in the communica�tions channel. In this case, the input signal of the neu�ron receiver is expressed as

Iin = Iout + η(t). (4)

Here, Iout is the output signal of the neuron transmitterand η(t) is the Gaussian noise with a zero mean anddifferent variances. Let us investigate the receptionquality at various levels of noise in the communica�tions channel. For certainty, it is assumed that theOOK scheme is applied to the receiving�transmittingsystem based on neuronlike elements. The receptionquality is estimated by calculating the bit�error proba�bility for a received signal at different levels of noise.The presence or absence of a spike is determined with thehelp of a threshold device. The threshold value is selectedso as to ensure the minimum probability of erroneousreception of a signal. In our case, the 80�mV threshold ischosen. The pulse width corresponding to the thresh�old is ~ 2 ms. Simulation has revealed that the spikedelay can be 2–3 ms at the high noise level. Such adelay can also be caused by the neuron inertia. Thus,

the time interval of spike location is 2–7 ms from thebeginning of each period. It is in this interval that thealgorithm determines the presence or absence of aspike. The performed simulation has made it possibleto obtain the dependence between bit�error probabil�ity Perr and the bit energy�to�noise density (Eb/N0)ratio (Fig. 7). This figure indicates that the neuron

300

200

100

0

–100

2001801601401201000 80604020–200

150

100

50

0

2001801601401201000 80604020–50

I, µA/cm2

(а)

(b)

t, ms

150

100

50

0

2001801601401201000 80604020–50

V, mVt, ms

t, ms

Fig. 6. Neuronlike receiver response to Gaussian noise: (a) Gaussian noise with the maximum amplitude on the order100 µA/cm2 and (b) the neuronlike receiver membrane potential.

100

10–1

10–2

10–3

–2 0 2 4 6 8 10 12 14

Еb/Т0, dB

Perr

Fig. 7. Bit�error probability Perr during spike reception vs.the Eb/N0 ratio.

434

JOURNAL OF CJOURNAL OF COMMUNICATIONS TECHNOLOGY AND ELECTRONICS Vol. 55 No. 4 2010

DMITRIEV, RYZHOV

receiver recognizes information transmitted with a bit�error probability of 10–3 at the bit energy�to�noisedensity ratio of about 13 dB. These data are compara�ble to the probabilistic characteristics of known inco�herent receivers.

CONCLUSIONS

The question about digital information transmis�sion between neuronlike elements has been discussed.The transmission system structure, which involves arectangular�pulse stream former, a neuron transmit�ter, a neuron receiver, and a threshold device, is pro�posed. It is shown that such a system is capable oftransmitting binary information and has the certainlevel of noise immunity. For the used parameters, the

transmission rate is about 100 bit/s. This value can beregarded as the rough estimate of data transmissionrate in living neuron systems.

REFERENCES

1. A. L. Nodgkin and A. F. Huxley, J. Physiol. 117, 500(1952).

2. J. L. Hindmarsh and R. M. Rose, Proc. R. Soc. London221 (1222), 87 (1984).

3. R. Fitzhugh, Biophys. J. 1, 445 (1961).4. G. D. Abarbanel, M. I. Rabinovich, A. Sel’verston,

et al., Usp. Fiz. Nauk 166, 363 (1996).5. J. G. Nicholls, A. R. Martin, B. G. Wallace, and P. A. Fuchs,

From Neuron to Brain (Sinauer Associates, Sunderland,Mass., 2001; Editorial URSS, Moscow, 2003).