Deterministic Prediction and Chaos in Squid Axon Response

Transcription

1 Deterministic Prediction and Chaos in Squid Axon Response A. Mees K. Aihara M. Adachi K. Judd T. Ikeguchi SFI WORKING PAPER: SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent the views of the Santa Fe Institute. We accept papers intended for publication in peer-reviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for papers by our external faculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, or funded by an SFI grant. NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensure timely distribution of the scholarly and technical work on a non-commercial basis. Copyright and all rights therein are maintained by the author(s). It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may be reposted only with the explicit permission of the copyright holder. SANTA FE INSTITUTE

2 Deterministic Prediction and Chaos in Squid Axon Response A.Mees(1), K.Aihara(2), M.Adachi(2), K.Judd(2), T.Ikeguchi(3) and G.Matsumoto(4) (l)mathematics Department, The University of Western Australia, Nedlands, Perth 6009 (2)Department of Electronic Engineering, Tokyo Denki University, 2-2 Kanda, Chiyoda, Tokyo 101 (3)Department of Applied Electronics, Science University oftokyo, Yamazaki, Noda 278 (4)Electrotechnical Laboratory, Umezono, Tsukuba 305, Japan Abstract We make deterministic predictive models of apparently complex squid axon response to periodic stimuli. The result provides evidence that the response is chaotic (and therefore partially predictable) and implies the possibility of identifying deterministic chaos in other kinds of noisy data even when explicit models are not available. 1

3 Deterministic chaos is often characterised by its long-term unpredictability. Short term prediction may, however, be possible if a good enough model is available, together with sufficiently accurate measurements. Recent work on building mathematical models directly from data[1-7] has made it possible to detect deterministic structure in some time series data which appear at first sight to be relatively unpredictable. Since biological systems are essentially nonlinear and nolllequilibrium[8 10), it is natural to expect existence of deterministic chaos in such systems. Various studies have been directed toward discovering and understanding chaos in biological systems such as brains and hearts[1l-13]. For example, it has been experimentally confirmed that when squid giant axons are forced periodically, the nerve membranes fire irregulady under some cir.cumstances[14 17]. Fig.1 shows a time series in which the membrane potential is stroboscopically sampled at the leading edge of each stimulating pulse. No clear regularity is apparent. The spectrum is shown in Fig.2: in spite of some peaks, it is not indicative of a periodic signal, or of a simple superposition of periodic signals. It does not exclude the possibility that this axon response could be stochastic. Because of non-stationarity due to axon fatigue, the data, which are in effect from a Poincare section[18-21)' are not readily available in sufficient quantity to allow reliable calculation of such statistics as fractal dimension. Estimation of the latter is, in any case, a delicate procedure, and even a reliable estimate with confidence intervals[22] will not necessarily distinguish deterministic from stochastic response. If an explanation for the system's behaviour is proposed-for example, in the form of a dynamical system-it can be tested by requiring it to make predictions and then validating or falsifying them by experiment. The present investigation adopted this approach. The Hodgkin-Huxley system[23] is the obvious candidate, and is an excellent mathematical model even for chaotic response[14,18-21], although it does not fit squid a..xon response well under certain conditions[24]. However, we wanted to have a method that would also work in other cases where no generally accepted model is available. We chose instead to create an approximate model system directly from the data. The model is a function (usually defined via a computer algorithm) which estimates the value of the membrane potential at time t + k, given its values at times 0, 1,...,t. Several methods are available[1-7]; we selected the tesselation[5,7] and neural network[3,25] approximations. 2

4 Both methods assume that the data has been embedded[26]. Figure 3( a) shows an embedding in two dimensions; that is, a plot of Xl+! algainst Xl' This is strong evidence that the data is not stochastic; the points lie almost on a one-dimensional curve, and certainly do not densely fill an area of the plane as a simple random process would. As the time series data seem to be well embedded in the two-dimensional space, we used an embedding dimension of 2 and a lag of 1 in the subsequent analysis. The time series data of the squid axon response were di vided into first and second halves for learning and testing, respectively. In deterministic prediction via tesselation[5,7], the first 250 points Xl, X2,.., X250 from our 500 point data set were embedded in the two-dimensional space and tesselated as the starting model for prediction. For each successive time t > 250, data up to and including Xl were used to make predictions i l + k of the response Xl+k for k = 1,2,3,4 where k is prediction time. In deterministic prediction via neural networks, the first half of the data was used for learning with the back-propagation algorithm[25]. The structure of the neural network was feed forward with three layers, having respectively 2 input, 9 hidden and 2 output neurons. The input signals to the first layer and the teaching signals to the last layer were (Xl, Xl+!) and (Xl+!' Xt+2), respectively. For t > 250, the output of the second neuron in the last layer gives one-step predictions. By feeding back the output signals to the input layer k times successively, we can also get k + I-step predictions. With either prediction method we can allow free-running to see whether an apparent strange attractor is produced. A possible strange attractor produced by the neural network model is shown in Fig.3(b). The result of the deterministic prediction is shown in Fig.4 where the ordinate is the correlation coefficient between actual and predicted values and the abscissa is the number of steps ahead being predicted. The solid and dotted lines in Fig.4(a) correspond to the tesselation predictions and the neural network predictions, respectively. The performance of the backpropagation network depends upon the initial choice of its connection weights and thresholds as demonstrated in Fig.4(b); the dotted line in Fig.4(a) shows the average performance over the five networks in Fig.4(b). The one-step prediction it+! was excellent in both methods, with correlation better than 92% between actual and predicted values. Correlations decreased as prediction time k increased, as shown in Figure 4. As argued by Sugihara and May[6], tltis is evidence for chaotic dynamics: chaos is charac- 3

5 terized by sensitivity to initial conditions, and the errors both from modeuing and from measurement will cause divergence between the predicted and the true val ues as prediction time increases. The Lyapunov spectrum[27] is a useful index of sensitive dependence on initial conditions. The largest exponent AI of the Lyapunov spectrum can be estimated from relationships between the prediction time T and statistical quantities such as the root-mean-square error E [2,28] and the correlation coefficient l' [29]. Using the relationship between T and E for iterated prediction given by Casdagli et al. [28], we obtain estimated values for AI of 0.35 for the tesselation and 0.36 for the averaged neural nehvork model of Fig.4(a). Using Wales's result (equation (7) in Ref.(29]) relating r to AI and T, the corresponding values of AI are 0.35 and 0.37 for the tesselation and the neural network, respectively. Moreover, the value of 0.39 is obtained by the Sano-Sawada algorithm[30]. These similar values give credence to the existence of a positive Lyapunov exponent in the dynamics of the squid axon response. In all our tests there was good agreement between the two qui te different modelling methods, so it seems unlikely that the results are a lucky accident. An alternative approach, which constructs a simple explicit model from understancling of refractoriness and other neural characteristics, also gives similar results[21]. Acknowledgements AIM thanks Tokyo Denki University for hospit.ality and JSPS and the Australian Academy of Sciences for a travel grant. This research was partially funded by ARC grant A and by a Grant-in-Aid( ) for Scientific Research on Priority Areas from the Ministry of Education, Science and Culture of Japan. A. I. Mees thanks the Santa Fe Institute and the Los Alamos National Laboratory. References [1] J.P. Crutchfield & B.S. McNamara, Comple:v Systems 1, (1987). [2] J.D. Farmer & J.J. Sidorowich, Phys. Rev. Letters 59, (1987). 4

6 [3] A. Lapedes & R. Farber, Nonlineal' signal pl'ocessing using neural netwol'ks: pl'ediction and system modelling (Los Alamos National Laboratory, 1987). [4] M. Casdagli, Physica D 35, (1989). [5] A.I. Mees, in Dynamics of Complex Intel'connected Biological Systems (eds. T. Vincent, A.I. Mees & L.S. Jennings) (Birkhauser, Boston, 1990). [6] G. Sugihara & R.M. May, Natul'e 344, (1990). [7J A.I. Mees, Intemational Joul'nal of Biful'cation and C/wos 1, in press (1991). [8] P. GlansdorIT & I. Progogine, Thel'modynamic theol'y of stl'uctul'e, stability and fluctuations (Wiley-Interscience, London, 1971). [9J G. Nicolis & I. Prigogine, Self-ol'ganization in nonequilibl'ium systems: f1'om dissipative stl'uctul'es to ol'del' th1'ough fluctuations (Wiley Interscience, London, 1977). [10] H. I-Iaken, Synel'getics - an Introduction: Transitions and Self- 0l'ganisation in Physics, (Springer-Verlag, Berlin, 1977). Nonequilibl'ium Phase Chemistr'y and Biology [I1J A.V. Holden, Chaos (Manchester University Press, Manchester, 1986). [12] H. Degn, A.V. Holden & L.F. Olsen, Chaos in Biological Systems (Plenum Press, New York, 1987). [13] B.J. West, Fractal Physiology and Chaos in Medicine (World Scientific, Singapore, 1990). [14] K. Aihara & G. Matsumoto, in Chaos (eds. A.V. Holden) (Manchester University Press, Manchester, 1986). [15] IC Aihara, T. Numajiri, G. Matsumoto & M. I<otani, Physics Letters A 116, (1986). [16] G. Matsumoto et ai., Physics Letters A 123, (HI87). 5

7 [17] N. Takahashi et a!., Physica D 43, (1990). [18] K. Aihara, G. Matsumoto & Y. Ikegaya, J. Theor. BioI. 109, (1984). [19] K. Aihara & G. Matsumoto, in Chaos in Biological Systems (eds. H. Degn, A.V. Holden & L.F. Olsen) (Plenum Press, New York, 1987). [20] K. Aihara, in Bifurcation phenomena in nonlinear systems and theory of dynamical systems (eds. H. Kawakami) (World Scientific, Singapore, 1990). [21] K. Judd, Y. Hanyu, N. Takahashi & G. Matsumoto, to be submitted to J. Math. Bioi. (1992). [22] IC Judd & A.I. Mees, Inte1'7lational Journal of Bifurcation and Chaos 1, in press (1991). [23] A.L. Hodgkin & A.F. Huxley, J. Physiol. (London) 117, (1952). [24] Y. Hanyu & G. Matsumoto, Physica D, in press (1991). [25] D.E. Rumelhart, G.E. Hinton & R.J. Williams, Nature 323, (1986). [26] F. Takens, in Dynamical Systems and Turbulence (eds. D.A. Rand & L.S. Young) (Springer, Berlin, 1981). [27] I. Shimada & T. Nagashima, Progress in Theoretical Physics 61, 1605 (1979). [28] M. Casdagli, D. des Ja.rdins, S. Eubank, J.D. Farmer, J. Gibson, N. Hunter & H. Theiler, Los Alamos National Laboratory, Report LA-UR (1991). [29] D.J. Wales, Nature 350, (1991). [30] M. Sano & Y. Sawada, Phys. Rev. Lett. 55, (1985). 6

8 Figure Captions Fig.l Squid axon membrane potentials stroboscopically sampled at the leading edge of each stimulating pulse. The squid axon in the resting state was stimulated by periodic pulses with amplitude 1.19 times the threshold current, pulse width 0.3 msec and pulse interval 3.8 msec. The temperature was 14.0'0. The number of data points is 500. The data are normalized between -0.5 and 0.5. Fig.2 Power spectrum of the squid axon response time series in Fig.l, computed with a Hanning window of width 64. The dashed lines are!~5% confidence intervals. Fig.3 (a) An embedding in 2 dimensions of the first half of the squid data in Fig.1. (b) A possible strange attractor obtained by allowing a ba.ck-propagation network modelling the squid data to free-run (that is, to simulate the system for many time steps). Fig.4 Correlation coefficient between actual and predicted values as a function of increasing prediction time. (a) The solid and dotted lines correspond to predictions using tesselation and using a back-propagation neural network, respectively. The dotted line is the average over five networks shown in (b). The initial values of connection weights and thresholds in each network of Fig.4(b) were determined randomly from a uniform distribution between -0.3 and 0.3. The neural nets were trained for a fixed time; on termination, the root mean square error between output and teaching signals in the last layer was in all cases less than

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