{"title": "Statistical and Dynamical Interpretation of ISIH Data from Periodically Stimulated Sensory Neurons", "book": "Advances in Neural Information Processing Systems", "page_first": 993, "page_last": 1000, "abstract": null, "full_text": "Statistical and Dynamical Interpretation of ISIH \n\nData from Periodically Stimulated Sensory Neurons \n\nJohn K. Douglass and Frank Moss \n\nDepartment of Biology and Department of Physics \n\nUniversity of Missouri at St. Louis \n\nSt. Louis, MO 63121 \n\nAndre Longtin \n\nDepartment of Physics \nUniversity of Ottawa \n\nOttawa, Ontario, Canada KIN 6N5 \n\nAbstract \n\nWe interpret the time interval data obtained from periodically stimulated \nsensory neurons in terms of two simple dynamical systems driven by noise \nwith an embedded weak periodic function called the signal: 1) a bistable \nsystem defined by two potential wells separated by a barrier, and 2) a Fit(cid:173)\nzHugh-Nagumo system. The implementation is by analog simulation: elec(cid:173)\ntronic circuits which mimic the dynamics. For a given signal frequency, our \nsimulators have only two adjustable parameters, the signal and noise intensi(cid:173)\nties. We show that experimental data obtained from the periodically stimu(cid:173)\nlated mechanoreceptor in the crayfish tail fan can be accurately approximated \nby these simulations. Finally, we discuss stochastic resonance in the two \nmodels. \n\n1 INTRODUcnON \n\nIt is well known that sensory information is transmitted to the brain using a \ncode which must be based on the time intervals between neural firing events or \nthe mean firing rate. However, in any collection of such data, and even when \nthe sensory system is stimulated with a periodic signal, statistical analyses have \nshown that a significant fraction of the intervals are random, having no coher(cid:173)\nent relationship to the stimulus. We call this component the ''noise\". It is clear \n\n993 \n\n\f994 \n\nDouglass, Moss, and Longtin \n\nthat coherent and incoherent subsets of such data must be separated. Moreover, \nthe noise intensity depends upon the stimulus intensity in a nonlinear manner \nthrough, for example, efferent connections in the visual system (Kaplan and \nBarlow, 1980) and is often much larger (sometimes several orders of magnitude \nlarger!) than can be accounted for by equilibrium statistical mechanics (Denk \nand Webb, 1992). Evidence that the noise in networks of neurons can dynami(cid:173)\ncally alter the properties of the membrane potential and time constants has also \nbeen accumulated (Kaplan and Barlow, 1976; Treutlein and Schulten, 1985; \nBemander, Koch and Douglas, 1992). Recently, based on comparisons of inter(cid:173)\nspike interval histograms (ISIH's) obtained from passive analog simulations of \nsimple bistable systems, with those from auditory neurons, it was suggested that \nthe noise intensity may play a critical role in the ability of the living system to \nsense the stimulus intensity (Longtin, Bulsara and Moss, 1991). In this work, it \nis shown that in the simulations, ISIH9s are reproduced provided that noise is \nadded to a weak signal, i.e. one that cannot cause firing by itself. All of these \nprocesses are essentially nonlinear, and they indicate the ultimate futility of \nsimply measuring the 'background spontaneous rate\" and later subtracting it \nfrom spike rates obtained with a stimulus applied. Indeed, they raise serious \ndoubts regarding the applicability of any linear transform theory to neural prob(cid:173)\nlems. \n\nIn this paper, we investigate the possibility that the noise can enhance the ability \nof a sensory neuron to transmit information about periodic stimuli. The present \nstudy relies on two objects, the ISIH and the power spectrum, both familiar \nmeasurements in electrophysiology. These are obtained from analog simulations \nof two simple dynamical systems, 1) the overdamped motion of a particle in a \nbistable, quartic potential; and 2) the FitzHugh-Nagumo model. The results of \nthese simulations are compared with those from experiments on the mechanore(cid:173)\nceptor in the tailfan of the crayfish Procambarus clarkii. \n\n2 THE ANALOG SIMULATOR \n\nPreviously, we made detailed comparisons of ISIH's obtained from a variety of \nsensory modalities (Longtin, Bulsara and Moss, 1991) with those measured on \nthe bistable system, \n\n.:t = x - x 3 + ~(t) + f sin(wt) \n\n(1) \n\n(~(t)~(s\u00bb = (DI r)exp( -\n\nwhere f is the stimulus intensity, and ~ is a quasi white, Gaussian noise, defined \nby \nIt-sll r) with D the noise intensity and r a \n(dimensionless) noise correlation time. Quasi white means that the actual noise \ncorrelation time is at least one order of magnitude smaller than the integrator \n\n\fStatistical & Dynamical Interpretation of ISIH Data from Periodically Stimulated Sensory Neurons \n\n995 \n\ntime constant (the \"clock\" by which the simulator measures time). It was shown \nthat the neurophysiological data could be satisfactorily matched by data from \nthe simulation by adjusting either the noise intensity or the stimulus intensity \nprovided that the other quantity had a value not very different from the height \nof the potential barrier. Moreover, bistable dynamical systems of the type rep(cid:173)\nresented by Eq. (1) (and many others as well) have been frequently used to \ndemonstrate stochastic resonance (SR), an essentially nonlinear process \nwhereby the signal-to-noise ratio (SNR) of a weak signal can be enhanced by \nthe noise. Below we show that SR can be demonstrated in a typical excitable \nsystem of the type often used to model sensory neurons. This raises a tantaliz(cid:173)\ning question: can SR be discovered as a naturally occurring phenomenon in \nliving systems? More information can be found in a recent review and work(cid:173)\nshop proceedings (Moss, 1993; Chialvo and Apkarian, 1993; Longtin, 1993). \n\nThere is, however, a significant difference between the dynamics represented by \nEq. (1) and the more usual neuron models which are excitable systems. A \nsimple example of the latter is the FitzHugh-Nagumo (FN) model, the ISIH's of \nwhich have recently been studied (Longtin, 1993). The FN model is an excit(cid:173)\nable system controlled by a bifurcation parameter. When the voltage variable is \nperturbed past a certain boundary, a large excursion, identified with a neural \nfiring event, occurs. Thus a detenninistic refractory period is built into the \nmodel as the time required for the execution of a single firing event. By con(cid:173)\ntrast, in the bistable system, a firing event is represented by the transition from \nwell A to well B. Before another firing can occur, the system must be reset by \na reverse transition from B to A, which is essentially stochastic. The bistable \nsystem thus exhibits a statistical distribution of refractory periods. The FN \nsystem is not bistable, but, depending on the value of the bifurcation parameter, \nit can be either periodically firing (oscillating) or residing on a fixed point. The \nFN model used here is defined by (Longtin, 1993), \n\nv = \\-(v - 0.5)(1 - v) - w + ~(t), \nw = v- w - [b + fsin(wt)], \n\n(2) \n\n(3) \n\nwhere v is the fast variable (action potential) to which the noise ~ is added, W is \nthe recovery variable to which the signal is added, and b is the bifurcation par(cid:173)\nameter. The range of behaviors is given by: b >0.65, fixed point and b ~ 0.65, \noscillating. We operate far into the fixed point regime at b = 0.9, so that \nbursts of sustained oscillations do not occur. Thus single spikes at more-or(cid:173)\nless random times but with some coherence with the signal are generated. A \nschematic diagram of the analog simulator is shown in Fig. 1. The simulator is \nconstructed of standard electronic chips: voltage multipliers (X) and operational \n\n\f996 \n\nDouglass, Moss, and Longtin \n\nOutput, P(T), P(w) \n\nPC \nAsyst \n\nsoftwo.re \n\nAction potential: fast variable \nv - v(v-O.5)(t-v) - w + Sit) \n\nS (,:.Lt. _---, \n+ n Noise \ngen. \n\nv(l-v)(f +0.5) \n\n+ g (t) \n\nn \n\nvet) \n\nv-w \n\nwet) \n\nRecovery: slow variable \nw = v-w-b- E sinwt \nb=0.1 to 1.0 \nv-w-b \n\nSignal \ngen. \n\nEsinwt \n\nFig. 1 Analog simulator of FitzHugh-Nagumo model. The characteris(cid:173)\ntic response times are determined by the integrator time constants as \nshown. The noise correlation time was Tn = 10-5 S. \n\namplifier summers (+). The fast variable, vCt), was digitized and analyzed for \nthe ISIH and the power spectrum by the PC shown. Note that the noise correla(cid:173)\ntion time, 10-5 s, is equal to the fast variable integrator time constant and is \nmuch larger than the slow variable time constant. This noise is, therefore, col(cid:173)\nored. Analog simulator designs, nonlinear experiments and colored noise have \nrecently been reviewed (Moss and McClintock, 1989). Below we compare data \nfrom this simulator with electrophysiological data from the crayfish. \n\n\fStatistical & Dynamical Interpretation of ISIH Data from Periodically Stimulated Sensory Neurons \n\n997 \n\n3 EXPERIMENTS WITH CRAYFISH MECHANORECEPTOR CELLS \n\nSingle hair mechanoreceptor cells of the crayfish tailfan represent a simple and \nrobust system lacking known efferents. A simple system is necessary, since we \nare searching for a specific dynamical behavior which might be masked in a \nmore complex physiology. In this system, small motions of the hairs (as small \nas a few tens of nanometers) are transduced into spike trains which travel up \nthe sensory neuron to the caudal ganglion. These neurons show a range of \nspontaneous firing rates (internal noise). In this experiment, a neuron with a \nrelatively high internal noise was chosen. Other experiments and more details \nare described elsewhere (Bulsara, Douglass and Moss, 1993). The preparation \nconsisted of a piece of the tailfan from which the sensory nerve bundle and \nganglion were exposed surgically. This appendage was sinusoidally moved \nthrough the saline solution by an electromagnetic transducer. Extracellular \nrecordings from an identified hair cell were made using standard methods. The \npreparations typically persisted in good physiological condition for 8 to 12 \nhours. An example ISIH is shown in the upper panel of Fig. 2. The stimulus \nperiod was, To = 14 ms. Note the peak sequence at the integer multiples of To \n(Longtin, et ai, 1991). This ISIH was measured in about 15 minutes for which \nabout 8K spikes were obtained. An ISIH obtained from the FN simulator in the \nsame time and including about the same number of spikes is shown in the lower \npanel. The similarity demonstrates that neurophysiological ISIH's can easily be \nmimicked with FN models as well as with bistable models. Our model is also \nable to reproduce non renewal effects (data not shown) which occur at high fre(cid:173)\nquency and! or low stimulus or noise intensity, and for which the first peak in \nthe ISIH is not the one of maximum amplitude. \n\nWe turn now to the question of whether SR, based on the power spectrum, can \nbe demonstrated in such excitable systems. The power spectrum typically \nshows a sharp peak due to the signal at frequency wo, riding on a broad noise \nbackground. An example, measured on the FN simulator, is shown in the left \npanel in Fig. 3. This spectrum was obtained for a constant signal intensity set \njust above threshold and for the stated external noise intensity. The SNR, in \ndecibels, is defined as the ratio of the strength S(w) of the signal feature to the \nnoise amplitude, N(w), measured at the base of the signal feature: SNR = 10 \n10glOS! N. The panel on the right of Fig. 3 shows the SNR's obtained from a \nlarge number of such power spectra, each measured for a different noise inten(cid:173)\nsity. Clearly there is an optimal noise intensity which maximizes the SNR. \nThis is, to our knowledge, the first demonstration of SR based on the power \nspectra in an excitable system. Just as for the bistable systems (Moss, 1993), \nwhen the external noise intensity is too low, the signal is not \"sampled\" fre(cid:173)\nquently enough and the SNR is low. By contrast, when the noise intensity is too \n\n\f998 \n\nDouglass, Moss, and Longtin \n\nIii 27.0 \n--..... \n.::! 24.0 \n~ 21.0 \n..... \n:g 18.0 \nC1J \nC1 15.0 \n>-\n:!:! 12.0 \n...... \n;; 9.00 \nIt! .g 6.00 \nt-\no. 3.00 \n\nIii 27.0 \n--..... \n.::::! 24.0 \n.... \n~ 21.0 \n:g 18.0 \nC1J \nC1 15.0 \n>-\n:!:! 12.0 \n.... \nE 9. 00 \nIt! .g 6.00 \nt-\no. 3.00 \n\n16.0 32.0 48.0 64.0 80 .0 96 . 0 112. 128 . 144.xEb60. \n\nTime \n\nImsl \n\n, \n\n16.0 32.0 48.0 64.0 80.0 96.0 112. 128. 144'xEb60. \n\nTime \n\nIms) \n\nFig. 2. ISIH's obtained from the crayfish stimulated at 68.6 Hz \n(upper); and the FN simulator driven at the same frequency with b = \n0.9, Vnoise = 0.022 V nns' and Vsig = 0.53 V nns (lower). \n\nhigh, the signal becomes randomized. The occurrence of a maximum in the \nSNR is thus motivated. SR has also been studied using well residence time pro(cid:173)\nbability densities, which are analogous to the physiological ISIH's (Longtin, el \nai, 1991), and was further studied in the FN system (Longtin, 1993). In these \ncases, it is observed that the individual peak heights pass through maxima as the \nnoise intensity is varied, thus demonstrating SR, similar to that shown in Fig. 3, \nbased on the ISIH (or residence time probability density). \n\n4 DISCUSSION \n\nWe have shown that physiological measurements such as the familiar ISIH pat(cid:173)\nterns obtained from periodically stimulated sensory neurons can be easily mim(cid:173)\nicked by analog simulations of simple noisy systems, in particular bistable sys-\n\n\fStatistical & Dynamica1lnterpretation of ISIH Data from Periodically Stimulated Sensory Neurons \n\n999 \n\n1200 ,---- - - - - - - - , \n\n1000 \n\n8.00 \n\n~66t:r:.~~~ ~ \n~ \n~ \n\n~~ \n\n~ \n~ \n~ 600 : ~ \n'\" \n\n4.00 \n\n- ~ \n\n.050 \n\n.100 \n\n.150 \n\n.250 \n\n.200 \n.300 \nFrequency (Kllz) \n\n.350 \n\n.400 \n\n.450xEIfiOO \n\nOO~OO I~~~~~.UO \n\nNOISE VOI.TACE (toIV.msl \n\n2 .00 \n\n>-\n~ 7 33 \nen \nc: \n\n\u2022 \n\n~ ~:D~ \n.... \nc... 2.20 \n~ 1.47 \nC1 \ntn \n\nt .733 \nx \no \nn. \n\nFig. 3. A power spectrum from the FN simulator stimulated by a 20 \nHz signal for b = 0.9, f = 0.25 V and V,.oise = 0.021 V nns (left). \nThe SNR's versus noise voltage measured in the FN system showing \nSR at V,.olse ~ 10 mV nns (right). Similar SR results based on the ISIH \nhave been obtained by Longtin (1993) and by Chialvo and Apkarian \n(1993). \n\ntems for which the refractory period is strictly stochastic and excitable systems \nfor which the refractory period is deterministic. Further, we have shown that \nSR, based on SNR's obtained from the power spectrum, can be demonstrated for \nthe FitzHugh-Nagumo model. \nIt is worth emphasizing that these results are \npossible only because the systems are inherently nonlinear. The signal alone is \ntoo weak to cause firing events in either the bistable or the excitable models. \nThus these results suggest that biological systems may be able to detect weak \nstimuli in the presence of background noise which they could not otherwise \ndetect. Careful behavioral studies will be necessary to decide this question, \nhowever, a recent and interesting psychophysics experiment using human in(cid:173)\nterpretations of ambiguous figures, presented in sequences with both coherent \nand random components points directly to this possibility (Chialvo and \nApkarian, 1993). \n\nAcknowledgements \n\nThis work was supported by the Office of Naval Research grant NOOOI4-92-J-\n1235 and by NSERC (Canada). \nReferences \n\nBemander, 0, Koch, C. and Douglas R. (1992) Network activity determines \nspatio-temporal integration in single cells, in Advances in Neural Information \nProcessing Systems 3; R. Lippman, J. Moody and D. Touretzky, editors; \nMorgan Kaufmann, San Mateo, CA. 43-50 \n\n\f1000 \n\nDouglass, Moss, and Longtin \n\nBulsara, A., Douglass, J. and Moss, F. (1993) Nonlinear Resonance: Noise(cid:173)\nassisted information processing in physical and neurophysiological systems. \nNav. Res. Rev. in press. \n\nChialvo, D. and Apkarian, V. (1993) Modulated noisy biological dynamics: three \nexamples; in Proceedings of the NATO ARW on Stochastic Resonance in Phy(cid:173)\nsics and Biology, edited by F. Moss, A. Bulsara, and M. F. Shlesinger, 1. Stat. \nPhys. 70, forthcoming \n\nOenk, W. and Webb, W. (1992) Forward and reverse transduction at the limit of \nsensitivity studied by correlating electrical and mechanical fluctuations in frog \nsaccular hair cells. Hear. Res. 60, 89-102. \n\nKaplan, E. and Barlow, R. (1976) Energy, quanta and Limulus vision. Vision \nRes. 16, 745-751 \n\nKaplan, E. and Barlow, R. (1980) Circadian clock in Limulus brain increases \nresponse and decreases noise of retinal photoreceptors. Nature 286, 393 \n\nLongtin, A. (1993) Stochastic resonance in neuron models, in Proceedings of \nthe NATO ARWon Stochastic Resonance in Physics and Biology, edited by F. \nMoss, A. Bulsara, and M. F. Shlesinger, J. Stat. Phys. 70, forthcoming \n\nLongtin, A, Bulsara, A and Moss F. (1991) Time interval sequences in bistable \nsystems and the noise-induced transmission of information by sensory neurons. \nPhys. Rev. Lett. 67, 656-659 \n\nMoss, F. (1993) Stochastic resonance: from the ice ages to the monkey's ear; \nin, Some Problems in Statistical Physics, edited by G. H. Weiss, SIAM, Phila(cid:173)\ndelphia, in press \n\nMoss, F. and McClintock, P.V.E. editors (1989) Noise in Nonlinear Dynamical \nSystems, Vols. 1 - 3, Cambridge University Press. \n\nTreutlein, H. and Schulten, K. (1985) Noise induced limit cycles of the Bonho(cid:173)\neffer- Van der Pol model of neural pulses. Ber. Bunsenges. Phys. Chern. 89, \n710. \n\n\f", "award": [], "sourceid": 594, "authors": [{"given_name": "John", "family_name": "Douglass", "institution": null}, {"given_name": "Frank", "family_name": "Moss", "institution": null}, {"given_name": "Andr\u00e9", "family_name": "Longtin", "institution": null}]}