Biological Cybernetics

Biol. Cybern. 47, 71-76 (1983)

0 Springer-Verlag 1983

An Analogue Simulation of the Luminosity-Channel in the Vertebrate Cone Retina 2. Frequency Analysis Robert Siminoff Institut fiir Physiologic der Freien Universifiit Berlin, Berlin

Abstract. A frequency analysis was performed on an analogue model of the vertebrate cone retina. Bode plots of magnitude and phase angle shifts were obtained from the various stages of the Model. While the cones were non-linear, the rest of the Model was linear and results could be explained by cascading of lowpass filters with lLinear summation of antagonistic inputs of differing time constants. The cut-off frequency of the Model was determined by the first stage, which was the cone pedicle, while the slope of the db magnitude versus frequency curve increased for each "synaptic delay" in the pathway. An initial increase in the magnitude of the output voltage at any given stage occurred up to the cut-off frequency and was due to the antagonistic input voltages of differing time courses. Physiological data were discussed in terms of the Model.

1. Introduction A previous paper (Siminoff, 1983) dealt with a description of the hardware for an analogue simulation of a model of the vertebrate cone retina developed from a synthesis of the relevant literature (Siminoff, 1980). The response characteristics of this analogue simulation were consistent with a linear model, if the photodiode (or cone) stage was by-passed. The roles of negative feedback from the horizontal cell (HC) to the cones, electrical coupling of HCs and the interplay of center and surround field input voltages to the bipolar cells (BC) were investigated and found to be highly significant for the development of receptive fields of BCs and, thereby, subsequent neurons. The present paper deals with a frequency analysis of the analogue model of the vertebrate cone retina using sinusoidal input voltages. Bode plots of magnitudes and phase angles will be presented and corn-

pared with the experimental data of the literature. The results are best described in terms of cascading lag units (used to simulate synaptic delays) with linear interaction of antagonistic inputs. The output voltage of the BC was the resultant of its center and surround field input voltages. Negative feedback and electrical coupling were shown to alter the balance between the 2 antagonistic inputs to the BC, thus altering the response characteristics of the BC. Physiological correlates could be explained in terms of the mechanisms built into the Model, such as negative feedback and the interaction of antagonistic inputs with differing lag times (or synaptic delays).

2. Testing Procedures To test the frequency response characteristics of the Model, a Solatron Model 1310 Frequency Analyzer was used. As with the use of square wave voltages in the previous paper (Siminoff, 1983), the cones were bypassed and the sinusoidal voltage output of the Frequency Analyzer was fed directly to the input stage of the cone pedicle (PC) (Fig. 1). Also one input to each

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Fig. 2. Bode magnitude and phase angle plots for the PC-Summator. The PC-Summator output voltage has zero delay and is therefore frequency independent. Solid lines and broken lines connect points representing values obtained without and with electrical coupling, respectively. The magnitude of negative feedback (fdb) from the HC to the PCs was determined by the setting of a 10 turn potentiometer and these settings are indicated under fdb in Table 1. A setting of zero represents no negative feedback while a setting of 10 represents the maximum amount of negative feedback Fig. 3. Bode magnitude and phase angle plots for the PC-Integrator. Symbols and abbreviations are defined in the text and the previous figure. The PC-Integrator output voltage has one delay and therefore is frequency dependent. Slopes are given in db/decade

of the 7PCs was connected to the sine wave output voltage of the Frequency Analyzer in order to simulate whole field illumination. The potentiometers controlling the magnitudes of negative feedback and surround field input voltages to the BCs were used to alter these parameters. Attenuations of voltage magnitudes and phase angles were determined at the various output stages of the electronic model for frequencies ranging from 0.1 to 1000 Hz. The sine wave input voltage to the PC-Summator stage was set at 3.16 V R.M.S. to insure that the working range of the operational amplifiers was not exceeded. However, use of the complete range of voltages from 0.01 to 10V R.M.S. gave the same results. 3. Results of Frequency Analysis A full description of the Model has been presented (Siminoff, 1983) and the terminology and abbreviations used in the present paper are the same.

3.1. Cone Pedicle The simulated cone pedicle (PC) receives its input voltages from the 7 simulated cones of the unit hexagon (UH). The sinusoidal output voltage of the Frequency Analyzer was introduced directly to the input stage of the PC (Fig. 1). The PC-Summator output voltage has no "synaptic delay" (i.e. no leaky integrator) and with no negative feedback or electrical coupling present, there were no attenuations of the input sine wave voltage at the output of the PCSummator nor were there any phase angle shifts with increasing frequencies (Fig. 2). Negative feedback has been shown (Siminoff, 1983) to decrease the magnitude of the output voltage of the PC-Summator and this decrease was proportional to the magnitude of the negative feedback. Electrical coupling, which increased the output voltage of the HC, by itself had no effect on the output voltage of the PC-Summator, but did potentiate the effects of negative feedback by a factor

73 of over 4 (Siminoff, 1983). The magnitudes of the output voltages of the PC-Summator as measured at 1 Hz confirm these previous findings (Fig. 2). With no negative feedback, the attenuation of the input voltage was - 4.7 db, reflecting the PC-Summator gain of 0.24. Addition of electrical coupling increased the magnitude of the attenuation to -6.2 db. Negative feedback produced attenuation of the magnitude of the output voltage as measured at 1 Hz, which was proportional to the magnitude of the negative feedback (Fig. 2), and electrical coupling potentiated this effect of negative feedback. The effects of negative feedback were decreased with increasing frequencies so that by 100 Hz the magnitude of the output voltage of the PC-Summator stage was back to the level measured with no negative feedback (i.e. -4.Tdb). Electrical coupling greatly potentiated the effects of negative feedback on the db magnitudes and phase angles (Fig. 2). With increasing amounts of negative feedback, particularly when potentiated by electrical coupling, an apparent resonance peak appeared between 30 and 40 Hz. Negative feedback produced a phase angle shift in the PC-Summator and the phase angle was back to its level as measured with no negative feedback by 100 Hz. The amount of phase angle shift introduced by negative feedback varied with the magnitude of the negative feedback with an apparent maximum value of 200 ~ Above 100 Hz the PC-Summator acts as if no negative feedback was present. The output of the PC-Integrator (Fig. 1) represents either the center field input voltage to the tonic BCs or the input voltage to the HC. Without negative feedback or electrical coupling, the magnitude of the output voltage of the PC-Integrator was - 4 . S d b (Fig. 3) as compared to -4.7 db for the PC-Summator (Fig. 2). The effects of negative feedback and its potentiation by electrical coupling on the magnitude of the output voltage of the PC-Integrator follows the same course as for the PC-Summator: thus, the output voltage of the PC-Integrator reflects the output voltage of the PC-Summator with a time lag introduced by the Leaky Integrator as well as phase angle shifts. Up to 10Hz the db magnitude remained relatively unchanged and with higher frequencies there was a rapid attenuation at - 19 db/decade with a cut-off frequency of 15 Hz (as measured as the corner frequency). This cut-off frequency was independent of the magnitude of negative feedback even when potentiated by electrical coupling. The apparent resonance peak, when present, introduced distortions of the cut-off frequency and slope. Without negative feedback the phase angle shift of the output voitage of the PC-Integrator was 90 ~ when compared to the phase angle shift for the PCSummator. Negative feedback produced a variable amount of phase angle shift at 1 Hz, which was back to

the level measured with no negative feedback by 100Hz. Negative feedback introduced a maximal phase angle shift of about 200 ~. 3.2. Horizontal Cell The amplitude of the HC output voltage was potentiated by electrical coupling (Fig. 4), as found in the previous paper (Siminoff, 1983). Negative feedback decreased the db magnitude as measured at 1 Hz (Fig. 4) and electrical coupling potentiated this effect. The HC output voltage also represents the surround field input voltage to the BC, but as the resultant of convergence of the 6-surround-HCs, has a cut-off frequency of between 13 and 14Hz with a slope of - 36 db/decade. The values for the cut-off frequency or slope were not altered by negative feedback even when potentiated by electrical coupling (Fig. 4). No apparent resonance peaks were observed at the output stage of the HC. The maximum phase angle shift was 180~ and negative feedback did not introduce any additional phase angle shifts. 3.3. Bipolar Cell The output voltages of the tonic BCs represent the algebraic sum of the center and surround field inputs voltages as modified by negative feedback and electrical coupling (Fig. 1). The interaction between the 2 antagonistic input voltages controls the magnitude of the output voltage of the BC (Fig. 5). With increasing frequencies there is an initial increase in the magnitude of the output of the BC producing an apparent resonance peak. Then there is a sharp cut-off frequency and slope. With either the center or surround field input voltage dominating, the curves are fairly flat up to the cut-off frequency and then there is a steep slope (Fig. 5). The cut-off frequency as measured was about 15 Hz. The phase angle shift was about 180~ when only a center field input voltage was present and when the surround field input voltage dominated, the phase angle shift was about 270 ~. 3.4. Amacrine Cell The output of the phasic amacrine cell (AC) represents the algebraic sum of its inputs, i.e. the 2 opposing types of BCs (Fig. 1). The Bode plot of magnitudes (Fig. 6) reflects this summation of 2 antagonistic inputs. With a pure HPBC input voltage, the db magnitude was 16 db at 1 Hz, while the addition of an input voltage from the DPBC produced a decrease in the amplitude of the output so that when the DPBC input voltage dominated, the db magnitude was about - 10 db. The cutoff frequency was about 19Hz with a slope of - 56 db/decade. Introduction of an input voltage from

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Fig. 4. Bode magnitude and phase angle plots for the output voltage of the HC. Abbreviations and conventions are as before Fig. 5. Bode magnitude and phase angle plots for the output voltage of the BC. The output voltage of the BC represents the algebraic sum of the antagonistic input voltages from the center and surround fields plus an additional delay (Fig. 1). The magnitude of the center field input voltage was kept constant while the magnitude of the input voltage from the surround field was varied by the setting of a 10 turn potentiometer. A setting of zero represents a pure center field input voltage while a setting of 10 represents a mixture of center and surround field input voltages with the surround field input voltage dominating. The center and surround field input voltages were approximately equal at a setting of 1.75. The magnitude of negative feedback (fdb) was 0.37 Fig. 6. Bode magnitude and phase angle plots for the phasic AC. The amount of HPBC input voltage was kept constant and the magnitude of the DPBC input voltage was determined by the setting of a 10 turn potentiometer. A setting of 4.25 represents equal amounts of HPBC and DPBC input voltages to the phasic AC. Negative feedback (0.37) and electrical coupling were present

the DPBC produced a phase angle shift of about 180 ~ while the phase angle shift was 200 ~ when the HPBC input voltage dominated.

3.5. Effects of Time Constants The effects of varying the RCs were studied for the first 3 stages of the electronic model, i.e. the PC-Summator with no lag unit, the PC-Integrator with one lag unit a n d the output voltage of the HC with 2 l a g units (Fig. 1). Table 1 tabulates the effects of varying the RC on the cut-off frequencies and slopes when no negative feedback was present so that the curves were reasonably flat up to the cut-off frequencies. The PCSummator stage is unaffected by increasing the RC. The other 2 stages (representing the input voltages to the BCs from the center and surround fields) both

showed decreases in cut-off frequencies with increasing RCs and have essentially the same values. Slopes were unaffected by the changing values of the RC, but increased with the number of lag units (or synapses). 4. D i s c u s s i o n

The present results using sinusoidal input voltages are consistent with previous results (Siminoff, 1983) obtained with square wave input voltages; the analogue model remains linear even with the introduction of electrical coupling of HCs, negative feedback from the HC to the PC and interaction of antagonistic input voltages with differing time courses. The electronic simulation performs faithfully the properties of the theoretical model (Siminoff, 1980).

75

4.1. Phase Angle Shifts The results are readily interpretable as a linear system of cascading low-pass filters (Fig. 1). The lag unit introduces a phase angle shift of 90~ this shift is maximal at the low frequencies with attenuation at the higher frequencies and a cut-off frequency dependent on the time constant of the Leaky Integrator. The Bode phase angle plots are completely consistent with these well-known properties of a linear system. The PC-Summator output voltage has no phase angle shift (Fig. 2) except when negative feedback is present, in which case a maximum of 180 ~ or 2 lag units are present (Fig. 4). The PC output voltage adds an additional lag unit producing a phase angle shift of 90 ~ without negative feedback and up to a maximum of 270 ~ with negative feedback (Fig. 3). The output voltage of the BC represents the algebraic sum of its antagonistic input voltages modified by a Leaky Integrator (Fig. 1). With only a center field input voltage present, the BC output voltage has 2 cascading lag units (Fig. 1) or a phase shift of 180 ~ (Fig. 5). A pure surround field input voltage would give a BC output voltage an additional phase angle shift of up to a maximum of 270 ~. Mixtures of center and surround field input voltages (0.5 to 3.0 of Fig. 5) give phase angle shifts intermediate to these 2 values dependent on which input voltage dominates. The phasic AC has 2 antagonistic input voltages with the HPBC input voltage having an extra lag unit. Thus, the output voltage of the phasic AC will have either an additional 90 ~ or 180 ~ phase ~ingle shift, depending on whether the DPBC or HPBC input voltage dominates (Fig. 6).

4.2. Magnitude Attenuations The Bode'magnitude plots are also consistent with a model of a system of linear interactions. Negative feedback attenuates the PC-Summator,output voltage (Fig: 2), an effect potentiated 4-f01d. by, electrical coupling (Figs. 2 and 4). The output, voltage of the BC rePresents the interabtion of i t s afitagonistic input voltages modified by negative feedback (Fig. 5). The addition of a surround field input voltage to the BC initially produces a further attenuation of the BC output voltage, but with further additions of surround field input voltages the output voltage of the BC becomes positive, indicating a dominance of the surround field over the center field. Addition of negative feedback attenuates either the effects of center or surround field input voltages (Fig. 5). With mixtures of center and surround field input voltages to the BCs, there are complex interplays due to the electrical coupling of HCs. Electrical coupling leads to a decrease in the output voltage of the BC when the center field is activated but an increase in the output voltage

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of the BC when the surround field is activated (Siminoff, 1980, 1983). The intermediate curves of Fig. 5 are consistent with such interplay. The addition of an antagonistic input voltage with a longer lag time, such as negative feedback, or surround field input voltage to the BC or HPBC input voltage to the phasic AC introduces an apparent resonance peak to the Bode magnitude plot (Figs. 2, 3, 5, 6). The antagonistic input with its extra delay has a lower cut-off frequency so that the initial increase in magnitude represents the loss of the antagonistic input voltage with increasing frequencies. The resonance peak (10-30 Hz) represents the region where the main stage of the system begins attenuation of its output voltage. The cut-off frequency is determined by the RC of the PC stage (Table i) and in the case of the retina this would be the cones. The slopes of the Bode magnitude plots are multiples of - 1 9 d b / d e c a d e (Table 1), at least up to the input stage of the BC. The output voltages of the BCs and the AC have more complex slopes, usually demonstrating an initial steep slope followed by a shallower slope of - 20 db/decade, thus reflecting the interplay of antagonistic input voltages of differing time constants.

4.3. Comparison of Model with Data of Literature The work of Abernethy (1974) has a definite relevance to the present study. His work concerned modelling of a 2-synapse feedback loop in the bipolar-amacrine pathway. He showed that the involvement of 2 first order processes in a feedback pathway led to an underdamped linear second order system in which resonance could occur with-the amount of resonance dependent on the amoun't of negative feedback. The appearance of resonance peaks with increasing amounts of negative feedback was seen in my Model (Fig. 2). There have been relatively few papers dealing with a frequency analysis of pre-GC units. The earliest paper is that of Spekreijse and Norton (1970) dealing with S-potentials in the isolated carp retina; the HC is the source of the S-potentials (Kaneko, 1979). Their

76 Bode magnitude plots are similar to those I obtained with my Model in a number of ways, except for one important difference. For the carp, the db magnitude was flat until about 7 Hz (as compared to 4 Hz for my simulated HCs) with a cut-off frequency at about 10Hz (as compared to 12Hz for an RC of 9.4ms). They obtained a slope of - 2 4 d b / o c t a v e , which is equivalent to 4 poles, since each pole adds 6 db/octave. The slopes of my simulated HCs were - 2 0 db/decade (or about - 10 db/octave), which is equivalent to only 2 poles, as was to be expected from my model. Toyoda (1974), also working in the isolated carp retina, found slopes of - 18 db/octave or - 24 db/octave depending on the intensity of the light stimulus. Foerster et al. (1979) found slopes of - t8 db/octave in HCs of the cat retina. Thus, there seems to be a discrepancy of one or 2 poles or lag units between my model and the data of the literature. The source of this extra pole or 2 is probably the transducer mechanism of the photoreceptor. My model does not include this segment of the retina and no doubt there is some delay mechanism between the absorption of the photon and the polarization changes of the photoreceptor's membrane. For the carp retina, T o y o d a (1974) found that the Bode magnitude plots of HCs had cut-off frequencies of 2 Hz, but were 5 Hz when the light stimulus was 1.0log units greater. For carp cones (Toyoda, 1974), the Bode magnitude plots were flat up to about 1-4 Hz, although at the higher light intensity there was some enhancement of the amplitude within this range. The cut-off frequency was 5-10Hz, again dependent on light intensity. The Bode plots of T o y o d a (1974) for cones agree well with the Bode magnitude plots for the PC-Summator output voltage (Fig. 2), suggesting that there is some negative feedback to the cones in the carp retina and that the RC is greater than 9.4 ms but less than 47 ms. T o y o d a (1974) also studied BCs and ACs and the Bode magnitude plots are remarkably like those of Figs. 5 and 6. For BCs, an increase in the amplitude occurred at the low frequencies with an apparent resonance peak between 3-5Hz. The resonance peak is lost when the surround field is under steady illumination and the cut-off frequency is estimated to be 4 Hz. The loss of the resonance peak is to be expected when only center field illumination is presented. For phasic ACs, the Bode magnitude plots showed a large increase in magnitude up to about 2 H z and then there was a sharp decrease with a slope of - 2 0 d b / o c t a v e (my estimation), while my simulated AC has a - 2 3 db/octave slope. T o y o d a (1974) felt the need to explain his result in terms of a non-linear AC, however, his data are compatable with a linear model. Foerster et al. (1979a, b) found 3 types of HCs in the cat retina, one of which they attributed to a pure cone input (Hw). These HCs had flat Bode db magni-

tude plots from i to 20Hz which then showed a resonance peak between 4 0 - 6 0 H z and a slope of 18 db/octave. Due to the high modulation used (0.8) and large spatial stimulus, non-tinearities were very apparent. Their H n (which they attributed to a pure rod input) had simptier Bode magnitude plots with a flat portion up to about 2 H z and then a cut-off frequency of 18 Hz with a slope of - 18 db/octave. The H,, (which they attributed to a mixture of rod and cone inputs) had Bode magnitude plots most difficult to reconciliate with my Model. Up to 2-4 Hz there was a slight decline with a slope of - 4 db/octave and a cutoff frequency at about 4 0 H z with a slope of - 3 1 d b / o c t a v e . Of the 3 types of HCs, their H m appears most closely to my simulated HC. The large modulations used probably introduced non-linear distortions. A subsequent paper will deal with the physiological correlates in more detail. -

Acknowledgements. I wish to acknowledge the continued support and generosity of Professor Dr. O.-J. GriJsser, without which this work would not be possible. This work was supportedin part by a grant from the Volkswagen Foundation

References Aberuethy, J.D. : A dynamic model of a two-synapse feedback loop in the vertebrate retina. Kybernetik 14, 187-200 (1974) Foerster, M.H., Grind, W.A. van de, Griisser, O.-J. : The response of the cat horizontal cells to flicker stimuli of different area, intensity and frequency. Exp. Brain Res. 29, 367-385 (1977a) Foerster, M.H., Grind, W.A. van de, Grtisser, O.-J. : Frequency transfer properties of three distinct types of horizontal cells. Exp. Brain Res. 29, 347-366 (1977b) Kaneko, A.: Physiology of the retina. Ann. Rev. Neurosci. 2, 169-191 (1979) Siminoff, R. : Modelling of the vertebrate visual system. I. Wiring diagram of the cone retina. J. Theor. Biol. 86, 673-708 (1980) Siminoff, R. : An analogue model of the luminosity-channel in the vertebrate cone retina. 1. Hardware and responses to square wave voltages. Biol. Cybern. 46, 101-110 (1983) Spekreijse, H., Norton, A.L.: The dynamic properties of color-coded S-potentials. J. Gen. Physiol. 56, 1-15 (1970) Toyoda, J.-I.: Frequency characteristics of retinal neurons in the carp. J. Gen. Physiol. 63, 214-234 (1974)

Received: April 19, 1982

Dr. Robert Siminoff Naval Submarine Med. Res. Lab. Naval Submarine Base, Box 900 Groton, CT, 06349 USA