The messages in optic nerve fibers and their interpretation

Brain ResearchReviews,16 (1991) 135-149 a 1991 Elsevier Science Publishers B.V. All rigbts reserved. 0165-0173/91/$03.50

135

TDONIS oi65oi739i90127T BRESR 90127

The messages in optic nerve fibers and their interpretation Ken-ichi Naka and Hiroko M. Sakai Departmentof Oph~~l~~o#, New York UniversityMedicalCenfer, New York, NY 10015 (U.S.A.) (Accepted 19 March 1991)

Key wor&: Spike train; Anaiog signal; Point process; Wiener kernel; Cascade analysis; Retina; Ganglion cell; White noise

CONTENTS 1. Introduction

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135

2. Background

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135

3. Analyses ................................................................................................................................................................ 3.1. Wiener analysis ................................................................................................................................................ 3.2. Cascade analysis ............................................................................................................................................... 3.3. Wiener analysis applied to a point process ............................................................................................................

137 137 139 141

4. Implications

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145

5. Conclusions

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147

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148

6. Summary

Acknowledgements References

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1. INTRODUCTION

“The world is my representation.”

It was around the 1930’s that the idea of spikes as a carrier of info~ation in the nervous system emerged. Among those who formulated such an idea was Adrian of the Physiological Laboratory in Cambridge, England ‘. Hodgkin and Rushton I1 showed that the spread of current in a nerve fiber could be described by cable theory. Hodgkin and Huxley” laid the fo~~tio~ for the ‘ionic theory’ of spike generation. With the introduction of the patch technique, in combination with molecular biological m~ipulations, the study of the molecular mechanism of spike generation will surely make impressive progress. Spike discharges are a means of communication. For

Cot~tqm&nce: Ken-i&i Naka, PHL 821, Department York, NY 10016, U.S.A. Fax: (1) (212) 779-8755.

148 148

a student of neuron networks, the archetype being the central nervous system, the relevance of spike discharges rests not so much with the (ionic or molecular) mechanisms of spike generation, but with the way in which a neuron encodes and decodes ~fo~ation into a spike train so that information can be transmitted from point A to point B within a network both efficiently and reliably. This is the network mechanism. In the case of the authors of this particular article, points A and B are the retina and the tectum, respectively. 2. BACKGROUND

Hartline*, among others, showed that there were 3 classes of optic-nerve discharges or retinal ganglion cells, namely, ON-, OFF-, and ON-OFF cells. He found ‘units’ which increased their discharges at the ON-set, at the OFF-set, and at the ON- and OFF-set of a brief flash of light or in response to a pulsatile stimulus given in the

of Gpbthahnology,

New York University Medical Center, 550 First Avenue, New

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mental flash (superposed on a fixed mean luminance) can be varied to determine the magnitude of the incremental flash required to evoke just one single discharge, as in the threshold experiment of Dormer’ in which the Weber-Fechner relationship for the frog optic nerve discharges was observed. In all the cases we have described so far, it was the stimulus that characterized a response, that is to say, experimenter determined the appropriate stimulus required to evoke a maximal number of discharges or the stimulus with the minimal threshold. Spike discharges are used as an indicator of the efficacy of a stimulus, and so such experiments represent an experimenter-centered approach6. The experimenter chooses a type of stimulus that experience has shown to be most suitable for achieving the goal of his particular experiments. To date, such an approach has proved quite satisfactory. Gerstein and Kiang’ developed a technique for producing a post-stimulus-time (PST) histogram by taking advantage of then newly developed electronics. They were able to exploit a means of converting a point process into an analog signal by repeating the same stimulus many times, one example being shown in Fig. 1. This technique relied on the fact that timing of discharges was stochastic, i.e. there was jitter in spike firing and two identical stimuli did not produce two exactly identical spike trains. If the discharges are exactly related to the stimulus, the PST histogram will simply reproduce the same point process. Jitter in the firing or noise in the spike-generation process tends to be viewed as a universal phenomenon but this is not necessarily the case. One example has been shown by Korenberg et al. I6 who made simultaneous intracellular and extracellular recordings from a single retinal ganglion cell. A sinusoidal cur-

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They

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wave-form of the stimulus that produced a spike discharge by running the recorder in the reverse direction and using discharges (transformed into unitary pulses) as a trigger to accumulate the stimulus waveform that produced a spike discharge. Their method is the reverse or trigger correlation and has been used extensively by the Dutch school in their studies of auditory system6. An experimenter presents a ‘rich’ stimulus and a neuron network responds to that particular part of the stimulus that is of interest to the network. The development of this important technique, however, has limitations. Reverse

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Fig. 1. Spike discharges evoked by a sinusoidal current injected through an intracelhrlar electrode into a catfish retinal ganglion cell. Spike discharges were recorded by a tungtem electrode from the same ganglion cell. Al shows 3 sample recordings and A2 PST histograms of the spike and current stimuhts produced by the averaging of about 20 repetitions. In this case, there was jitter in the firing of discharges, which allowed the conversion of the discharges, a point process, into an analog form. Bl shows 3 sample recordings from another cell. B2 is a PST histogram made exactly as in A2. There was no jitter in the firing and the histogram is identical to single sweep recordings. From Korenberg et al.t6 reproduced with the permission from the American Physiological Society.

137 correlation shows us the optimal wave- form of the stimulus required to evoke a discharge but it is not possible to relate the result to neuronal processes in the network. In this sense, the reverse correlation still falls within the traditional framework of seeking a specihc stimulus to evoke a response. Nonetheless, introduction of the technique was significant because the technique has focused attention on the dynamics of the stimulus. Schellart and Spekreijse35 used this technique to study the dynamics of responses of color-coded ganglion cells in goldfish retina. They made a Fourier transform of the kernels (reverse correlograms) to represent the optimal stimulus waveform in the frequency domain as gain and phase. Victor et a1.3g and Victor and Shapley3* developed an analytical technique of using a sum of several sinusoidal signals to test a system. Their technique occupies the middle ground between the classical technique of testing a system by a series of sinusoidal signals and Wiener’s idea of testing a system by an all-inclusive white-noise input. Victor and Shapley cross-correlated a light stimulus, a sum of sinusoidal signals, with the spike discharge (transformed into unitary pulses) from the Y-ganglion cells of the cat. They obtained second-order kernels represented in the frequency domain, which led them to some interesting conclusions about the neuron network that is associated with the generation of spike discharges from X- and Y-cells. The efforts of Victor and Shapley were important because they were the first to show that a particular sequence of spike discharges, detected by a second-order correlation, carried relevant information. Victor and Shapley were also the first to introduce the notion of cascade structure, the sandwich/Korenberg model, to the network analysis. Interpretation of Victor and Shapley’s kernels in terms of neuronal responses remained, however, ambiguous, i.e. their kernels were not related directly to the network mechanism. Their kernels derived from the light stimulus, the input, and spike discharges, the output, could not be equated with the analog potentials in the ganglion or preganglionic cells. The maximum frequency of spike discharges rarely exceeds 500 Hz and is very much lower than the pulse rate (carrier frequency) of man-made communication devices that are currently running at a rate of MHz or GHz. Although the limitation is partly compensated by the large number of fibers or lines of communication, there must be an appreciable process for abstraction of information as well as for compression in the production of spike discharges to compensate for the lower rate of transmission. For example, it would be advantageous to encode information in a non-linear manner to compress information. A second-order non-linearity can be encoded by changes in the relative timing of two sequential spike discharges, and a third-order non-linearity can

be encoded by 3 sequential discharges, and so on. Here it is the pattern of spike discharges, the internal structure of each spike train, that is important and not other measures, such as the average frequency of firing, the rate of instantaneous firing or the number of spikes in a given time bin. To date, no analytical approach has been advanced to reveal such a structure, with the exception of the example provided by Victor and Shapely. In this article we will show that (1) the kernels obtained by a correlation between a light stimulus and spike discharges (transformed into unitary pulses) can be related to the ganglion cell’s PSP and (2) the second-order non-linearities are encoded into a spike train. 3. ANALYSES

3.1. Wiener analyses The functional identification of a system originated with two mathematicians, Volterra and Frechet, who, almost 80 years ago, attempted to describe the output of a finite memory system by a series of Volterra functionals (a functional is a function whose argument is a function and whose value is a number). Wiener was interested in chaotic phenomena and made, early in his career, important contributions to the mathematical description of Brownian motion14. Brownian motion, a fundamental chaos, is still the subject of rigorous mathematical analysislO and Gaussian white noise is a formal derivative of the Brownian motion. The idea of using a stochastic process or a chaotic input as a tool to probe a system must have been in the mind of the great mathematician for a long time. White-noise analysis was Wiener’s last major contribution to science. A radical idea may be conceived when the mind is young but it takes a mature mind to bring the idea to fruition. Wiener4’ suggested the use of a stochastic input, Gaussian white noise, to probe a system and to identify it through a series of (Wiener) kernels. Here we quote from Wiener”: “If a circuit is nonlinear, if, for example, it contains rectifiers or voltage limiters or other similar devices, the trigonometric input is not an adequate test input. In this case, a trigonometric input will not in general produce a trigonometric output. Moreover, strictly speaking, there are no linear circuits, but only circuits with a better or worse approach to linearity. The test input that we choose for the examination of nonlinear circuits - and it can be used for linear circuits, too - is of a statistic nature. Theoretically, unlike the trigonometric input, which must be varied over the entire range of frequencies, it is a single statistical ensemble of inputs that can be used for all transducers. It is known as the shot

138 effect. Shot-effect generators are well-defined pieces of apparatus with a physical existence as instruments, and may be ordered from the catalogue of several houses of electrical-instrument makers” Nowadays the shot-effect generator is called a whitenoise generator. Wiener analysis is usually performed on a system of which the input as well as the output are analog in form. Cross-correlation is made between two signals to produce a series of kernels”. If a system is linear the system’s first-order kernel is the system’s impulse response to an impulse, referred, therefore, to as the impulse response. An impulse is a mathematical notion and, in reality, an impulse-like input is used. If a system is non-linear, the first-order kernel is the best linear approximation of the system’s impulse response. This process of recovering a system’s impulse response by cross-correlation is shown in Fig. 2 in which a system is a Tektronix AF501 bandpass filter tuned at 35 Hz. We stimulated the filter with a white-noise signal and observed the output. Cuss-chelation was made between the input and output to produce the system’s first-order kernel. As the system is man-made the system’s impulse response can easily be observed by a pulse of very short duration (which approximates an impulse input). Two impulse responses, one obtained directly and the other through cross-correlation are similar. However, there are minor differences between two impulse responses. These differences may be due to several reasons such as the finite length of the white-noise stimulus and the deviation of the input white-noise from a true Gaussian white noise, i.e. the noise was necessarily band-cited. Now we replace the Tektronix bandpass filter with a catfish neuron network (Fig. 3). Input is a white-noise modulated current injected into an NA (depolarizing sustained) amacrine cell and a resulting response was

recorded from an NB (hyperpolarizing sustained) amacrine cell. The input signal was cross-correlated against the response to obtain a first-order kernel. We found that the transmission of signal from the NA to NB amacrine cell is quasilinear. The kernel approximates, therefore, the impulse response of neuron network between the NA and NB amacrine cells. The kernel from the catfish is very similar to that from the Tektronix bandpass amplifier. Both are characteristics of a 35 Hz bandpass titer. It is notewo~hy that both the NA and NB amacrine cells produce a spontaneous oscillation of about 35 Hz either in the dark or in the presence of a steady illumination13. Generation of 3.5 Hz spontaneous oscillation is probably related to the 35 Hz filter that exists between the two amacrine cells33. The second-order kernel is a 3-dimensional structure; two axes represent timing of two input impulses and the third axis the maguitude of second-order non-linearity produced by the interaction of two impulses. The diagonal value shows the amplitude of the non-linear interaction when two pulses arrive simultaneously, i.e. when the amplitude of a pulse is changed. Although a secondorder kernel can be shown as a perspective of a solid (Fig. 4A), we prefer contour representation (Fig. 4B). This kernel has a signature that we refer to as a “Ceye structure in which two positive peaks, shown by solid lines, and two negative peaks, shown by dashed lines, occupy 4 corners of a square. The second-order kernel is produced by a Wiener structure, a dynamic linear filter followed by a squaring device, shown in Fig. 6. ~eoreti~ally, first-order kernel can be measured by stim~a~g a system with an impulse input or a pulse that approximates it as we did in the case of the Tek-

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Fig. 2. impulse responses of a Tektronix AT501 bandpass filter tuned at 35 Hz. Two impulse responses are shown. The one in the upper box was produced by brief pulses aviating an impulse input and the one in the lower box was obtained by cross-correlation between a white-noise input and the fitter’s output.

Fig. 3. A first-order kernel obtained by cross-correlation between an input, a white-noise current injected into an NA amacrine cell, and a response recorded from an NB amacrine cell. The kernel identifies the signal tmnsmimion from the NA to NB amacrine cell, which was quasi-linear. The kernel was similar to that for the Tektronix AF501 bandpass filter tuned at 35 Hz. The signal transmission from ANOVA to NB amacrine cell is modeled by a bandpass filter tuned at 35 Hz as in the case of man-made Tektronix filter.

139 tronix bandpass filter. In reality, this is not always the best method for obtaining an impulse response. An impulse input large enough to produce an observable response may drive the system into a region of non-linearity. Similarly, second-order kernel can be measured with two impulses given at various intervals. For a seasoned experimenter this is a very tedious process. A whitenoise stimulus stimulates a system ~ntinuously whereas an impulse input can be given only once in a while, i.e. a white-noise stimulus is an efficient stimulus. If we are going to examine dynamics of a chemical plant, it is much more sensible as well as practical to use a smooth perturbation around the plant’s operative point rather than to introduce a sudden and impulse-like change in the plant. In testing a visual system, the amplitude of an impulse input is necessarilly limited as light has no negative value. Although this is purely a technical point, an impulse-like test signal is difficult to use in the case of

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Fig. 4. A typical second-order kernel displayed in two ways. A second-order kernel is a Sdimensional solid with two time axes that correspond to two input pulses, and the third dimension, the Z-axis, represents the magnitude of non-linear interaction. A: a bird’s eye view of the solid. Depolarization is shown as a positive peak and hyperpolarization by a negative valley. The kernel shows a deviation from a linear supexposition of respom evoked by two pulse inputs. In this case, pulses were brief flashes of light superposed on a mean luminance that corresponded to the mean of the white-noise light stimulus. %vo time axes represent the two pulses. El: a contour map representation of the second-order kernel. The diagonal line shows the non-linear interaction when two pulses are given simultaneously, i.e. when the amplitude of a pulse is doubled. As the timing of two pulses is interchangeable, the kernel is symmetric around its diagonal. Depolarization is shown by solid lines and hyperpolarization by dashed lines. The kemel has a characteristic signature that is referred as a ?-eye structure29*“. This signature shows that the nonlinearity was of the kind produced by a Wiener structure, a linear differentiating Nter followed by a square-law device.

two-electrode experiments in which a test current is injected through one intracelhdar electrode and the resultant response is recorded by the other intracellular electrode, an example being shown in Fig. 3. Difficulty arises because such a signal contains high-frequency components which produce a cross-talk between two electrodes. In our early development of the white-noise analysis for network analysis, we en~untered niceties in dealing with the spike discharges. Although kernels could be obtained by a reverse-correlation technique, they could not be related directly to the kernels from the pre-ganglionic cells as we have already discussed. Note in this regard that almost all signal processing in the vertebrate retina is achieved by means of an analog signal, a fact that made the vertebrate retina very amenable to Wiener analysis. (Although spike discharges were reported in some amacrine cells, the role of such discharges in signal processing in the retinal network is not clear.) In the ganglion cells, at the final stage of retinal processing, the analog signal is converted into a point process, spike discharges. Wiener analysis in its original form is applicable to a system with an analog input and an analog output. Marmarelis and Naka21 tried to circumvent the problem by producing a PST histogram obtained by multiple repetition of identical, pseudo-random white-noise stimuli. This approach was time-consuming and introduced an artificial ~~-~equency component during the process of the assignment of a spike discharge to a given bin. ‘I%e single sweep method described in this article contrasts with the method developed by Marmarelis and Naka in 1973. Earlier results of physiological applications of whitenoise analysis can be found in Marmarelis and MarmareIi? and some examples of recent applications are found in Citron and Emerson3 and Korenberg et al.15. 3.2. Cascade analysis In a linear system, components in the system are commutable. There is no clue, unless it is obtained by other means, that allows one to decide the sequential order of the components in the system. In a non-linear system ~m~nents are generally not commutable. This property has given rise to a branch of Wiener analysis called cascade analysis pioneered by Korenberg14 and Victor and Shapley%. There are 3 typical cascade structures: Hammerstein, Wiener and Korenberg. H~erstein structure is a cascade of NL-L, Wiener structure of L-NL, and Korenberg structure of L-&L-L, where L is a dynamic linear and NL is a (static) non-linear component or filter (Fig. 5). The Korenberg structure is also referred to as a ‘sandwich’ structure38. A typical Wiener structure is a dynamic linear filter that differentiates an input with subsequent introduction

140

of a squaring (square-law) device. This typical Wiener structure is the structure Schetzen36 used as an illustrative example of static non-linearity. The output of such a structure has no linear component and is represented by a second-order kernel with a characteristic signature we call ‘4- or 9-eye structure’29. The second-order kernel is produced by a two-dimensional multiplication of the system’s first-order kernel, the dynamic linear element. This process is papery shown in Fig. 6. The response from the C (transient) amacrine cell of the catfish is approximated by this structure, in which the output of the dynamic linear filter is possibly the response of the bipolar cells. In our scheme, in the C amacrine cells, the bipolar-cell responses are transformed into the ON-OFF responses through a process equivalent to a squaring. The process is instantaneous and represents a static non-linearity (because there is no time component in the operation). In some C amacrine cells, a secondorder model (predicted by the first- and second-order kernels) predicts the cell’s response to a white-noise input with a M.S.E. (mean square error) of 20-30%. The M.S.E. of the first-order model is more than 90%, i.e. the linear component comprises less than 10% of the total response. Thus, the C amacrine cell’s response is largely composed of the second-order component. Experimental results suggest that there are many (local) L-NL units and a light flash covering a large area produces a synchrouized response from many primeval units (distributed spatially) to produce an ON-OFF depolarization. A stimulus, such as a whirling windmill40 or a moving bar of light4 produced a sustained de~la~zation of a C amacrine cell because the unitary responses produced by the stimulus were asynchronous in time. There is another class of amacrine cells: N (sustained) amacrine cells. The cell’s modulation response is composed of linear as well as non-linear components. The

Hammerstein

linear component accounts for about 30%-60% of the total response (measured in terms of MSE) and the second-order component about 20%. The second-order component has a signature that is distinctly different from that of the C amacrine cell. Sakai and Naka3’ showed heuristically that some of the N cell’s second-order kernel could be reproduced by a high-pass operation performed on the C cell’s response. The process of generation of N cell’s second-order non-linea~ty can, therefore, be modeled by a cascade of L-NL-L in which the first L-NL is the Wiener cascade representing the C amacrine cell response and the second L represents the filter interposed between the C and N amacrine ceils. The second linear filter should be recovered by crosscorrelation between a white-noise current injected into a C cell and the resulting response obtained from an N cell. It is, therefore, possible to test experimentally the scheme we present here. The L-NL-L cascade is a Korenberg or sandwich structure. The cascade is graphically shown in Fig. 7. Our analysis shows that the C amacrine

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Fig. 6. Generation of a C second-order kernel. A white-noise input characteristically produced the illustrated response from a C amacrine cell. Cross-correlation between the input and the response produced a typical C second-order kernel. The same whitenoise was processed through a bandpass linear filter whose impulse response is shown. The output of the filter was squared, a static non-linear operation as it is an instantaneous operation. The squared output was cross-correlated with the original white-noise input to prcxfuce a second-order kernel, which is very similar to that from a catfish C amacrine cell. ‘Ihe bandpass filter followed by a square-law device is a Wiener structure. Reproduced from Sakai et al. 31 by permission.

141 cell is the sole source of the second-order static non-lin-

earity in the inner retina. The signature of the amacrine cells’ second-order non-linearity is very characteristic and it can be used to trace the flow of signals in the retinal neuron network (just as a detective traces a fugitive by the signature of his finger print). Indeed such signatures are traced from the amacrine cells to the spike discharges from the ganglion cells as shown in the first column in Fig. 9. It is fortuitous that the characteristic signatures are kept unchanged in the retinal neuron network. 3.3 Wiener analysis applied to a point process In the vertebrate retina spike discharges are generated in the ganglion cells. An intracellular recording from the cell consists of two components, a postsynaptic potential (PSP) and spike discharges generated when the amplitude of the PSP is above a threshold value. The former component is an analog signal and the latter a point process. Sakuranaga et al.% recorded intracellular re-

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sponses from a catfish ganglion cell evoked by a whitenoise modulated light stimulus. They decomposed the evoked response into two components: the analog PSP and spike discharges. Fig. 10 is an example of such an analysis. Cross-correlation between the white-noise stimulus and the resulting PSP generated the hrst- and second-order kernels and is an example of standard whitenoise analysis. Sakuranaga et al. made a similar correlation between the stimulus and the resulting spike discharges, transformed into unitary pulses. They discovered that two outputs, the PSP, which is an analog signal, and the spike discharge, which is a point process, produced almost identical first- and second-order kernels. Sakai and Naka29 compiled the hrst- and secondorder kernels from bipolar, amacrine and ganglion cells and compared them with those computed from spike discharges. (Responses were recorded under comparable conditions.) Comparison was made between the kernels from the ON, OFF, and ON-OFF cells. Sakai and Naka found that the first-order kernels from analog potentials were almost identical to those from spike trains (Fig. 8). They also recovered 3 distinct types of second-order kernels from spike discharges, which were very similar to those from C, NA and NB amacrine cells and to those from ON-OFF, ON-, and OFF-ganglion cells (Fig. 9).

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Fig. 7. Generation of an N second-order kernel. A white-noise input characteristically produced the response from an NB cell. Cross-correlation between the input and output produced a typical N second-order kernel. The same white-noise was processed through a bandpass tilter, whose impulse response is shown. The filter is identical to the one shown in Fig. 5. The tirst filter’s output was instantly squared. The squared output was passed through a second linear filter, whose impulse response is also shown. Crosscorrelation between the white-noise input and the output of the second linear filter produced a second-order kernel similar to the one from cattish N amacrine cell. Reproduced from Sakai et al. 31 by permission.

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Fig. 8. First-order kernels computed from analog signals from horizontal, bipolar, amacrine and ganglion cells. .Kemels computed from spike discharges, point processes, arealso shown. Responses from different cell types as well as different ceils recorded under similar conditions and 4 examples from 4 cells are shown for each cell type. BA and BB denote ON- and OFF-center bipolar cells. NA and NB denote NA and NB amacrine cells. GA and GB denote ON- and OFF-center ganglion cells. H denotes horizontal cells. First-order kernels from horizontal cells are also shown to serve as a reference. Reproduced with permission from Sakai and Nakam.

142 Sakai and Naka’s finding showed that the first- and second-order components generated by the pre-ganglionic cells, bipolar and amacrine cells, were ‘copied’ as the ganglion cell’s PSP. The PSP was then translated into spike discharges without any major modification. The observation that signals generated in the pre-ganglion cells were copied as the ganglion cell’s PSP is not surprising. Sakai and Naka32 showed a bidirectional transmission of a signal between N amacrine and ganglion cells of the same response polarity with respect to their response. These trans~ssions are fast and low-pass so that signals that originate in the pre-ganglionic cells are transmitted to the ganglion cells without any major transformation. Korenberg et a1.16 made simultaneous intra- and extra-cellular (spike discharges} recordings from a single ganglion cell. One example of such measurements is shown in Fig. 10 in which a white-noise modulated light was used to produce the cell’s response. The exact syn-

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chrony of the spike discharges in the intra- and extracellular recordings shows that responses were recorded from a single ganglion cell. The spike discharges from the intracellular recording can be removed by use of the procedure employed by Sakuranaga et a1.34. Applying this procedure, we separated the intracellular response into an analog component and a point process (spike discharge). We obtained two sets of kernels, one set computed from the analog component and the other set from the point process (Fig. 11). As already shown by Sakuranaga et al., two sets of kernels are very similar in their waveform, i.e. signature, but there were subtle differences between the two sets. For example, the peak time of the first-order kernel, which was computed from the point process was slightly shorter than that from the analog response. This subtle discrepancy can be explained if we assume that there is a filter between the analog potential and the spike generation mechanism. To character&e such a filter, a white-noise modulated current was injected through the intracellular electrode to evoke discharges from the ganglion cell (recorded by the extracellular electrode). We cross-correlated the input white-noise current with the evoked spike discharges (transformed into unitary pulses). We recovered a brief

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Fig. 9. Typical second-order kernels computed from analog signals from amacrine and ganglion cells and spike discharge. The kernels in column marked ‘type-C are from ON-OFF transient cells, C amacrine and ON-OFF (C) ganglion cells. The kernels have the typical ‘4eye’ structure shown in Figs. 2 and 4. They are Produced by a cascade of L-NL, a Wiener structure. The cohmm marked ‘type-NA’ shows the kernels from ON-center cells, NA amacrine and ON (A) ganglion cells. The column marked ‘type-NR’ shows kernels from an NB amacrine and an OFF (B) ganglion cell. Kernels for the ON and OFF ceils can be approximated by a high-pass operation performed on a 4-eye kernel (Pig. 8). The kernels for the ON and OFF cehs can be modeled by an L-NL-L cascade, a Korenberg structure. Reproduced by permission from Sakai and Nakass.

2

B3 = Cl c2 , 0.1s , Fig. 10. Siiultaneous recordings from a catfish retinal ganglion cell. Two electrodes were used, one was intracellular and the other was a tungsten extracellular electrode. Responses were evoked by a field of light that was modulated by a white-noise signal (A). Bl shows the response recorded by the intracellular electrode. The rcsponse was decomposed into two components, the analog PSP (B2) and the spike discharges transformed into unitary pulses (B3). Cl shows spike discharges recorded by the tungsten electrode and C2 shows the discharges transformed into unitary pulses. The exact synchrony of discharges in the intracellular and extracellular recordings show that two recordings were from the same ganglion dell. Record Bl was transformed into records B2 and B3 by application of the algorithm by Sakuranaga et al.“. Reproduced with permission from Korenberg et aL3r.

143 CORRELATION

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Fig. 11. The first- and second-order kernels computed from the PSP and spike discharges from recordings similar to the one shown in Fig. 8. Two first-order kernels are similar but subtle difference in their waveform is seen. The spike kernel has a shorter peak response time and is more differentiating. Similarly, two second-order kernels, Bl and B2, show subtle differences.

B

Fig. 13. Two-input white-noise experiment. ‘Rvo independent whitenoise stimuli stimulated receptive-field center, as a spot of light, and surround, as an annulus of light. The output is a train of spike discharges. Cross-correlation between the discharge and two inputs extracts the tirst- and second-order kernels from the center and surround responses. This is based on Mannarelis and Naka’s extension of Wiener’s original theory into multi-input systems**.

but well-defined first-order kernel (Fig. 12). The PSP kernels were convoluted with the kernel, marked h in Fig. 12, to obtain a new set of first- and second-order kernels. The new set of kernels was very similar to the kernels computed from the spike train. Thus, although the generation of spike discharges is approximated by a Wiener structure, the model can be refined to include a second linear filter. The revised model is an L-NL-L, a Korenberg, structure. The first L-NL structure, the Wiener structure, represents the process that leads from the absorption of photons by the photoreceptors to the generation of ganglion cell’s PSP. The second L represents the transfer function between the PSP and spike generation. It is possible to include in the scheme the static non-linear process generated by the amacrine CdlP.

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Fig. 12.~Transformation of ganglion cell’s slow-potential kernels into spike kernels. The kernels k, and ka show the same set of firstand second-order (slow-potential) kernels as already shown in Fig. 11. Kernel h shows the first-order kernel obtained by white-noise current injected through the intracellular electrode to evoke spike discharges from the same ganglion cell. Cross-correlation was made between the current and resulting spike discharges (recorded extracellularly). The kernel is a linear filter that existed between the ganglion-cell slow potential and spike generation mechanism. Kernels k, and k, were convoluted by kernel h to produce two kernels, k,*h and ka*h. The dashed line is the same spike kernel which is shown in Fig. 9A. Two kernels, one from the spike train and the other transformed from slow potential are almost identical. The transformed second-order kernel, ka*h, is also very similar to that from the spike tram shown in Fig. 9 B2. Reproduced by permission from Korenberg et al.“.

Marmarelis and Naka” extended Wiener’s original theory into a multi-input system. One simple example of such a system is the concentric receptive field which is stimulated simultaneously by a spot and an annulus of light. This is graphically shown in Fig. 13. The spot and annulus of light are modulated by two independent white-noise signals and resulting spike discharges are cross-correlated with the two input signals. The crosscorrelation recovers two first-order kernels, one for the receptive-field center, stimulated by the spot, and the other for the surround, stimulated by the annulus, from a single spike train (Fig. 14A). The two recovered kernels are opposite in their polarity and show similar dynamics. The kernel for the surround, however, has a longer latency, which suggests that the surround signal takes a complex pathway to reach the region where the spike discharges are generated, which is probably the field’s center. Similarly, two second-order kernels are

144 computed from the same trains for the receptive-field center and surround (Fig. 14B). Two kernels are similar having 4-eye signature already shown in Figs. 4 and 6. In Fig. 15, we used incremental and decremental steps as inputs and computed the linear and (second-order) non-linear models by the convolution of the step inputs by the first- and second-order kernels shown in Fig. 14. The two first-order kernels, one for the spot and the other for the annular response, produced the linear models shown in Fig. 15 Al and A2. The spot and annular stimuli produced responses (predicated by the first-order kernels) of opposing polarity indicating an antagonistic and concentric receptive field. The responses predicted by the two second-order kernels shown in Fig. 15 Bl and B2 are transient depolarizations, or increases in spike discharges, at the ON- and OFF-sets of the pulsatile stimulus. Because the second-order terms are quadratic, both the incremental and decremental steps produced identical responses. In both the linear and second-order models, the spot models have a shorter latency than the annular models. Results shown in Figs. 14 and 15 show that the con-

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centric field organization is encoded by the linear component and not by the ~cond-order component. The second-order kernel shown in Fig. 14 bear the signature that is characteristic to the C (transient) amacrine cells. The C amacrine cells produce a depolarizing, transient response at the ON- and the OFF-set of step modulation of a mean luminance. In the catfish retina, the C amacrine cells are electrically coupled, possibly through gap junctions” and form the C-space (Sakai and Naka, in preparation). The space is similar to the S-space formed by horizontal cells25. Teranishi et al.37 reported dye-coupling between goldfish transient amacrine cells (which correspond to the C amacrine cells in cattish). The C amacrine cells form a monophasic field and so do the second-order components of the ganglion-cell responses. As we have already shown, signals generated in pre-ganglionic cell are translated into a spike train without any major modification. A single train of spike discharges carries simultaneously, in our two-input (spot and annulus) experiments, 4 kinds of information on the cell’s concentric receptive field; two linear components carry information on the center and surround and two non-linear components carry information on changes in luminance taking place anywhere in the field. A two-input stimulus can be replaced by a much more complex spatio-temporal stimulus. In the catfish’ and monkey (Reed and Shapley, personal communication), linear components of a spatiotemporal receptive field have been

0.4

ANNULUS

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Fig. 14. Kernels recovered from a spike train generated by a cattish ganglion cell. ‘Iwo first-order kernels shown in A were for the receptive-field center (solid line) and surround (dashed line). %vo kernels are opposite in their polarity showing the biphasic (concentric) and antagonistic nature of the field. The spot kernel is hyperpolarizing and the annular kernel depolarising. The cell was an off-center ganglion cell. The annular kernel has a longer latency as well as a peak response time. ‘Iwo second-order kernels, one for the center (Bl) and the other for the surround (B2), are very similar in that their signature is Ceye. We note, however, that these two second-order kernels show subtle differences. ‘Ihe annular kernel is more low-passed (at a cut-off frequency of about 15 Hz) as indicated by the elongated peaks and valleys. The treason of signals from the surround to the center may involve a low-pass filtering. This figure shows that information on the spatial organization is carried by the linear component and information on changes anywhere in the field is carried by the second-order component.

.

Fig. 15. Model (predicted) responses for the receptive-field center and surround. Incremental and decremental steps were convoluted by the IIrst- and second-order kernels from the spot and anmdar responses. The models for the spot responses are in solid and those for the annular responses in dashed lines. The linear models predicted by the first-order kernels for the incremental and decmmenta1 steps were mirror images of one another, an observation expected in a linar system. The second-order models are identical for the incremental and decremental steps because the kernels are quadratic functions. This simulation study clearly indicates that the concentric receptive-field organization has to be encoded by the linear components whereas the second-order components signal sudden changes in lumiuance occurring anywhere in a receptive field.

145 recorded. The linear component was a time-lapse movie of the rise and fall of a concentric receptive field. We will discuss this topic in the next section. So far we have shown that the complex neural processes leading to the generation of spike discharges can be dissected into a series of cascade structure, a few examples of which we have represented so far. The fact that the generation of spike discharges can be approximated by a cascade structure greatly simplifies our analysis of neuron network in the retina. One example is recovery of the spatiotemporal receptive-field component just discussed. 4. IMPLICATIONS

Our results from various experiments have shown that the kernels obtained by cross-correlation between a white-noise input and spike discharges, transformed into unitary pulses, recovered the analog PSP generated in the ganglion cell. (The slight discrepancy can be accounted for by a second linear-filter as we have shown above.) This recovery is possible because the spike generation mechanism is highly non-linear but is static. Two sets of kernels, one from an analog and the other from a point process, are related to one another through a proportionality factor. The spike kernels have units of spikes/s, whereas the PSP kernels units of mV. If we are interested in the dynamics but not in the absolute magnitude of PSPs and if our interest is within a second-order approximation, it is not necessary to record intracellular responses from ganglion cells but recording of spike discharges will suffice. Our experience has shown that the ganglion cell’s response evoked by a white-noise light stimulus could be modeled by the first- and secondorder kernels with an MSE of 20-30%. With an (almost) noise-free response such as that from horizontal cells, MSE of less than 10% could be achieved. In the case of other retinal cells in which recording is much noisier, an MSE of about 20% is the best that can be achieved. The parts of the response which are not accounted for by the first- and second-order kernels may include several factors such as (1) the sharp comers or peaks which are due to the higher-order components and (2) noise in the response. It is, therefore, reasonable to assume that the essential part of a ganglion cell’s response can be predicted (modeled) by the first- and second-order kernels. This predictability is a great advantage because recording of spike discharges is routine and is far more stable than intracellular recording. Pharmacology of ganglion cells as well as that of preganglionic cells can be studied by observing kernels derived from spike discharges. In this brief review of spike discharges, we have used examples from catfish retina which are very familiar to

us. First- and second-order kernels very similar to those from catfish have been recovered from ganglion cells in &verse animals such as fish which showed color processing (Sakai and Naka, unpublished), fiog”, and rabbit (Mangel, Sakai and.Naka, in preparation). The conclusions drawn here may be applicable to retinas in general. We have also found that, in some fish, green and red stimuli and spot and annular stimuli produced 4 firstorder kernels that can be recovered from a single spike train. The second-order kernels carry information on changes and not on color or space. The polarity of the slow potentials in the pre-ganglionic cells is preserved in the spike first-order kernels, whereas in the second-order kernel information about the polarity is lost. The kernel is a quadratic function providing a good reason, we believe, for encoding information on space and color as a linear component and information on changes as a non-linear component. Spike kernels are computed from an input white-noise signal, light stimulus, and spike discharges transformed into unitary pulses. What the kernel extracts is the optimal input waveform for production of a spike discharge (although time runs in the opposite direction): a standard interpretation of kernels produced by a reverse correlation’. The kernels computed from PSP are the changes in the membrane potential produced by a light stimulus. We have shown that two sets of kernels, one computed from PSPs and the other from a point process, match with one another quite well. Thus the linal product of the signal processing in the retinal network appears in the ganglion cells as the optimal light stimulus to produce a spike discharge. Conversely, the spike-generation mechanism is such that it produces a spike discharge most efficiently when a given PSP is generated. As the spike discharges are the final output of retinal neuron network, the conclusion drawn here has an important bearing in dissecting the network. There is no a priori reason that this should be so, and the important coincidence may be by design or by accident, we don’t know which. However, it is only through this coincidence that the spike kernels can be integrated into the network function and to be equated with the analog potentials in the network. In a sense, the visual world of the catfish or a part of it which is of vital interest to the fish is represented in the spike trains. There is a practical side to our study, which is related to the economy of signal transmission. In our whitenoise experiments on spike trains, a test stimulus is usually given for about 40 s. If we assume an average spike firing rate of 200 per s, the total number of spike discharges recorded is 8ooO. The second-order kernel is a 3-dimensional structure (Fig. 4) and a kernel is computed up to 100 points, which correspond to a maximal

146 delay of 0.2 s. A second-order kernel is, therefore, represented by a matrix of 100~100x100 points (assuming that the amplitude is represented by 100 points). The total number of points which form a second-order kernel is 1,000,000. Here we ignore the fact that only a limited part of the kernel contains meanin~l information and the fact that the kernel is symmetric around its diagonal. A second-order kernel can be recovered from a spike train containing less than 10,000 spikes or the times of event. This is a compression of 1:lOO. If a retinal ganglion cell is used as a translation stage, a 3-dimensional image such as a topographic map can be sent by a point process which contains l/100 of the information in the original map (provided that the same whitenoise signal is available at both ends of the transmission line). This (overly) simplified and crude estimation illustrates two important functions of the retinal neuron network: the generation of second-order non-linearity and coding of the non-hearty into a spike train. The principal of information compression we found in the retinal neuron network can conceivably be used to produce an artificial machine that is capable of trans~t~g information very efficiently over a long distance. The efficiency relies on two facts, one is that what is transmitted is a point process and the other is the ‘richness’ of whitenoise signal that need not be transmitted. In the above discussion, we have restricted our description to the responses evoked by modulation of a field, a spot or an annulus of light, all very simple light stimuli. In the real world, input to the visual system is a scintillation of light in time and space. The best approximation of the real visual world is a spatio-temporal white noise. It is possible to produce a response from a ganglion cell by simplified spatiotemporal white noise. One example of such stimulus is the ‘snow’ seen on unused television channels’ and another is a computer-generated checker pattern, the individual elements of which are modulated by a white-noise or M-sequence signal (Ref. 23; Reed and Shapley, personal communication). Reverse correlation between the input and output, spike discharges being transformed into unitary pulses, produces a set of 3-dimensional pictures. The series of pictures is the best linear appro~ma~on of the spatial input (at each delay time) that most efficiently produces a spike discharge. Such a series of pictures can be interpreted either as a time-lapse movie of the STFU? (spatiotemporal receptive field) or as the optimal spatio-temporal stimulus for production of a spike discharge. Time runs in the opposite direction for these two interpretations. We have shown that the discharges from a concentric field yield (temporal) kernels which can be equated ta the analog potentials in the ganglion and preganglionic cells. The STRF kernels, on the other hand,

cannot be equated directly with a cell’s analog potential although the difference between a two-input stimulus and a spatiotemporal stimulus is a matter of complexity but not of kind. Elucidation of the relationship between the STRF kernels and the network function is an important topic for future study. Such a study, if successfully executed, will probably uncover the most important principle of network mechanism in the visual pathway. In our analyses, we have taken full advantage as an external observers who have knowledge on both the input, a white-noise signal, as well as the output, a spike train. We segregate a spike train evoked by a spot and a annular stimulus into two linear and two non-linear components through a process of cross-#rrelation. What we have actually shown is that the retinal output is efficient in representing kernels whereas for catfish the spike train is a means to represent the external world. And although we have shown kernels represent, in a limited sense, part of such a stimulus, this conclusion is far from general. The spike train by itself does not contain sufficient information to extract kernels but it is the ‘richness’ of white-noise stimulus (which is available to us but not to catfish) that enables us to reconstruct kernels as we have already discussed. Spike trains are sent to the tectum in catfish and the tectal neurons decode the trams. There are two possibilities on how the neurons process the spike trains. (1) The neurons in the tectum are interested, as are many neurophysiologists, only in the instantaneous frequency of discharges. The complex spike-train structure we have discovered is an artifact of our analysis, or is simply ignored by the tectal neurons. (2) A more reasonable assumption is that all the information carried by a spike train ( and recovered by us) is utilized by the neurons in the tectum. We don’t know how this is done. It is possible that correlation of PSPs generated in many tectum cells is involved. To perform such a function, it is very adv~tageous if spikes from many optic nerve fibers arriving at the tectum are synchronized. Then one spike train can be used as a reference to the other train and vice versa. Lately, a series of studies by Sakai and Naka32 have shown that neurons in the inner retina are extensively and bidirectionally coupled through a fast pathway, i.e. the on cells form an ON-cluster and OFF-cells an OFF-cluster. This network mech~ism will ensure that timing of spike generation in one cell can be synchronized to some degree with those from neighboring cells. Two ganglion cells do not receive exactly the same inputs from preg~~ioni~ cells as a stimulus is a scintillation of light in time and space. There must be certain differences between spike trains from two ganglion cells, although spikes from many cells tend to be synchronized. The difference can

147

be detected by a process similar to a correlation and used to reconstruct, so to speak, catfish’s visual world. This truistic possibility is, however, an artifact and it is not likely that caffish tectum neurons receive and process information in a fashion similar to our thought process, the Wiener analysis being one example. It is possible that the decoding of signals by the tectal neurons or even encoding of information in the retina may be carried out in a fashion that is not appreciable intuitively or cannot be easily visualized. The mechanism of coding and decoding is one of the most basic problems in the study of the central nervous system. In the past, attention has been narrowly focused on the type of stimulus to evoke discharges but not enough thought has been given on the kind of information carried by a spike train and on how it is decoded. This is apparently due to the fact that neurophysiology, particularly visual neurophysiology, is a visually oriented science and, unfortunately, extremely complex functions of neuron networks are often visualized by use of simplistic pictures. Proper appreciation of such a complex process may require abstract approaches similar to those found in other branches of science. The way to proceed might be to begin with theoretical analysis of experimental results just as physicists have done so successfully. Of course, experiments have to be performed within a framework of a theory if experimental results were to be analyzed based on the theory. A lot, we believe, has been missed in many current studies in which effects of pharmacological agents are accessed by observing simply an increase or decrease in the number of discharges or by measuring spike intervals. 5. CONCLUSIONS

In this review we have shown that there is a fine structure in the spike train. The structure carries information on the second-order (static) non-linearity generated in the amacrine cells. It would be difficult to characterize such a structure by the methods currently used to analyre a spike train (there are a few exceptions). Study of neuron network, as well as coding of information into a spike train, is more abstractive than the study of ions, channels, and transmitters but it leads to the heart of the problem; the principal function of a neuron network is signal processing. The role of spike discharges as a carrier of information was discovered by Lord Adrian more than half a century ago and is still an important but neglected area of study. The coding of spatiotemporal information into a spike train is one example and decoding of many kinds of information carried simultaneously by a single spike train is another. The study is not only an intellectual challenge, but is of practical importance

as the study may uncover new principles of signal processing leading to a more efficient way of communication. ‘Ihe ultimate goal of science is to relate natural phenomena that we observe to an abstract idea represented by mathematical relationships, or to discover an observable which correponds to an abstract idea. The Euclidian notion of a point or a line is the most prominent example. White-noise analysis is an example, though far less prominent, of an abstract idea based on the theory of stochastic process being related to an observable. It is inevitable that ‘tension’ arises when such a contact is made, that is, when an abstract theory is used to interpret experimental results. For example, the white noise used in our catfish research is not an idealixed one on which the theory is constructed, and, in the real world, rounding errors are bound to be introduced during computation and the estimate of the power level of an input is problematic, to name but a few examples of mismatches. Thus, it is not possible to compute rigorously Wiener kernels (J. Victor, personal communication). Nevertheless, as we have shown here, Wiener analysis has provided valuable insight into the network mechanisms in the catfish retina and the second-order kernel has proved to be essential in our study. It is often mentioned that it is hard to use Wiener kernels to specify physiological, network or even computational properties of a system under study. Answer to this comment is evident in this article. Mismatches between theory and experiment are expected in any pathfinding effort and should not discourage us from opening a new path. One aspect of science involves an attempt to reduce the degree of mismatch or tension between theory and experiment. Reduction of tension may be achieved by improvements in our experimental approach and by modification of the theory. Such attempts at incremental progress represent the most basic theme of scientific research. Empirical phenomena may be true or may not be true; only abstract ideas bear truth in an absolute sense. It is not possible to remove entirely one’s prejudice or maya, a mythical veil or tinted glasses that ‘colors’ our view. This is particularly the case when one deals with an extremely complex as well as multivariate system such as the neuron network. Only when we relate what we observe to rigorously proven abstract ideas, is our observation endowed with a degree of permanence as well as of catholicity. This is the very reason theory is important in sciences. Unfortunately, the mismatch tends to be far more serious in some areas of biological sciences, which include neuroscience, than in other sciences and gives some biologists misgivings with respect to the application of abstract theory. In spite of the serious mismatches, the application of Wiener’s theory to the anal-

148

ysis of spike train is unique, we believe, in an empirical science of neuroscience. “For theories constitute the network of coordinates for science” so wrote Poppe?. 6. SUMMARY

Spike discharges are the principal carriers of information in the nervous system. Although both the ionic and the molecular mechanisms of spike generation have been studied extensively, the methods for analyzing a spike train that are currently employed have not changed much from those in use 20 years ago. There is an apparent need for a refinement of the methods used to analyze spike trains. We present here a summary of our recent results of an analysis of spike trains from retinal ganglion cells that is based on Wiener’s theory of non-linear analysis or white-noise analysis. We found that spike trains carry, at least to a second-order approximation, as much information as is carried by the ganglion cell’s postsynaptic potential (PSP). There is no loss of information when an analog signal, PSP, is converted into a point process, namely, spike discharges. It is indeed possible

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to predict the cell’s PSP from a spike train. This finding has two important implications. First, the neuron network in the retina produces a PSP, the dynamics of which are optimal for triggering a spike discharge, or conversely, the spike-generation mechanism is optimized to match the dynamics of the network. The external stimulus that is optimal for production of a ganglion-cell discharge is represented as the cell’s PSP. Second, there is structure encoded within the spike train; information on a second-order non-linearity is encoded by the relative timing of two consecutive spike discharges. Coding of non-linearity into a spike train is an efficient means of signal compression and is an important aspect of neurophysiology.

Ackowledgements. We thank our friends in New York who kindly read and commented on the manuscript. Preparation of this manuscript has been supported by NE1 Grant 07738 and NSF Grants DIR 8718461 and BNS 891993. K.I.N. also thanks Research to Prevent Blindness for his Jules and Doris Stein professorship. Citation from God and Golem, Inc., by Nobert Wiener is by permission from the M.I.T. Press.

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