Quantitative modeling of responses ot anuran retina: Stimulus shape and size dependency

Vision Rex. Vol. 33, No. 16, pp. 2361-2379, 1993 Printed in Great Britain. All rights reserved

Copyright

0

0042-6989/93 $6.00 + 0.00 1993 Pergamon Press Ltd

Quantitative Modeling of Responses of Anuran Retina: Stimulus Shape and Size Dependency JEFFREY

L. TEETERS,*?

MICHAEL

A. ARBIB,*$ FERNANDO

CORBACHO,*

HYUN BONG LEE*

Received 24 June 1992; in revised form 22 March 1993

Teeters and Arbib presented a model of the anuran retina which qualitatively accounts for the characteristic response properties used to distinguish ganglion cell type in amuans. In this paper we test the model’s ability to reproduce quantitatively tabulated data on the dependency on stimulus shape and size, with a new implementation of the model in the neural simulation Language NSL. Data of Ewert and Hock relating toad R2, R3, and R4 ganglion cell responses to moving worm, antiworm, and square-shaped stimuli of various edge lengths are used to test stimulus shape and size dependency. A close match to the data can be achieved by tuning some of the model parameters while still retaining the characteristic responses to the typical stimulus types. We stress here the importance of a populational approach to the models. We place more emphasis on the variation of response properties in a population of neurons of the sume class, rather than questing for the neuron of a given type. As an example of the populational approach we offer a model for the respiratory R3 response following researchers who argue that a subclass of R3 neurons are activated by stationary boundaries owing to the anuran’s self induced respiratory eye movement. Anura

Retina

1. THE TEETERS

Neural

networks

Quantitative

modeling

AND ARBIB MODEL

anuran retina model of Teeters and Arbib (1991) (henceforth referred to as T&A) accounts for the qualitative characteristic response properties used to classify anuran ganglion cell types. In this paper we show that, with minor tuning, the model can also account for quantitatively determined ganglion cell responses dependent on stimulus size and shape. While full details of the T&A model are given in the aforementioned paper, for convenience we summarize the model here because it is a starting point for this paper. The structure of the T&A model is shown in Fig. 1. The top part shows the layers of cells which feed all the ganglion cells, while the bottom part shows the specific inputs for each ganglion cell type. Each single cell in these diagrams represents a layer of cells in the formal model. We summarize the equations for the T&A model in Table 1, and will present revised models as the exposition progresses. Receptors (R) convert light energy into neural potentials. The hyperpolarizing response to light is modeled by setting the receptor potential to the inverse of light intensity (i) which ranges from 0 (dark) to 1 (light). Adaptation and other complexities are not included in the model. The

*Center for Neural Engineering, University of Southern California, Los Angeles, CA 90089-2520, U.S.A. TPresent address: University of California at Berkeley, Neurobiology Group, Room 145 LSA, Berkeley, CA 94720, U.S.A. .jTo whom all correspondence should be addressed.

Populational

variability

Horizontal cells (H) form the surround receptive field of both bipolar cell types. They are modeled so that they are only sensitive to the background illumination of the surround (HO in Table 1) and are spatially invariant (uniform potential model) through the infinite spread of the activation within the cells. This simple interpretation of horizontal cell function ignores the effect of presentation of a local stimulus and suggests that their main function is to bias the bipolar cells so they operate in their region of maximal sensitivity. Bipolar cells (HBC, DBC) are computed as a difference between receptor and horizontal cell activity. Hyperpolarizing bipolar cells (HBC) hyperpolarize in response to light, depolarizing bipolar cells (DBC) depolarize in response to light. PBH and PBD are the positive components of the HBC and DBC responses. Transient amacrine cells (ATH, ATD) convert the sustained bipolar outputs into transient signals. The transient amacrines are modeled as pseudo differentiators which operate by subtracting the leaky-integrated bipolar potential from the sustained bipolar potential, and then amplifying the difference if it is above threshold. We modeled the bipolars and amacrines to have one-to-one connections from the preceding layers based on the following assumptions: (i) horizontal cells in this model have a uniform potential which in effect makes the spatial connection properties mostly irrelevant, and (ii) dendritic tree diameters of the bipolars and amacrines are smaller than those of the ganglion cells. Thus, the T&A model input to the ganglion cells (receptors through bipolar and amacrine) ignores optics,

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JEFFREY

L. TEETERS TABLE

A&rithm (r)

et al.

1

for T&A model base (receptors through amaadne cells) R=l-i

Receptor

(b) Horizontal

zH$

= HO - H

offchannel

! HO = 0 ambient light, 1 ambient dark; Q = 0.1 onchaunel

(4 Bipolars HBC=R-H

DBC=H-R

PBH = msxWEfC,

0)

PBD 3: max(DBC, 0)

(d) Amaaines d(HBX) -=HBc-HBX ‘a dt

d(DBX) -= ‘5a dt

!ra=03

ATH, = max(HBC-HBX, ATi&1. e -At/%q

DBC-DBX

AT4 = max(DBC-DBX, ATDt_~ - e-At/%,)

AlgorithmsforTdkAmodelof&m@oncells BO Cells

RO= kOIATD - kl * ((3. ATH) + ATD) with k0 = mask& 1.8, I), kl = mesk(15.5,3.7,0.8) Rl Cells Rl =

k0 * (PBD+PBH+ATD+ATH) - kl * (ATD+ATH)

with k0 = mask(3,2.3, l), kl = mask(19.5,4.6,3) IU Cells The model uses two temporary variables (tc and g). “tc” is the total transient input tc the cell. “g” is a gate which is set to 1 if the net transient excitation is larger than thf inhibition. tc = k&ATH - kl * (ATH+ATD) g = pos(tc)

where pas(x) = 1 if x > 0, 0 otherwise

R2 = g . ((kO*PBH)+ tc) with k0 = mask(4,2.4, 11, kl = masW19.5,4.6,3) R3 Cells a=p*ATD+ATH R3 = k&a - (kl*a)&@d with p x 0.4, k0 = mask(8,2.4,1), kl = mask(19.5,4.6,1.4) while (s)d&@ =

Signal

8 delayed

by 40 milliseconds.

R4 Cells R4=kO*(ATH-x.ATD) with x = 1, k0 = mask(15.5,3.5,1).

different receptor types, light adaptation, and distinctions between other subtypes of horizontal, bipolar, and amacrine cells. It is not our claim that this simplification exhausts the functionality of these cells. Rather, we seek to emphasize that only these properties analyzed in this paper are essential for understanding the range of ganglion cell properties described here. In fact, the T&A

implementation of the horizontal and bipolar cells does not really affect the outcome of the stimulus shape and size discrimination tasks. Nevertheless we need the horizontal and bipolar cells to account for other phenomena caused by changes in whole field illumination. Gang&n c&s (RO-R4) receive input from bipolar and amacrine cells. Unlike the bipolar and amacrine cells

which have one-to-one connections to their preceding layers, each ganglion cell input is composed of a central ERF (excitatory receptive field), and a wider IRF (inhibitory receptive field). The spatial properties of the ERF and IRF are specified as two-dimensional Gaussians. The notation “Mask(dia, sig, wgt)” in Table 1 denotes a two-dimensional Gaussian with standard deviation sig (in visual degrees) which is truncated with diameter dia (so that the Gaussian values are replaced by 0 for points more distant than dia/2, also in visual degrees from the center), and which is normalized so that the sum of all elements is equal to wgt (for a more detailed description see the Methods section). The ERF extent is modeled as arising from ganglion cell dendritic

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RETINA MODEL

tree topology which is narrowly spread, whereas the IRF arises from a more widely spread topology. Some cell specific information about the T&A model ganglion cells will be given later in this paper as the tuned model is described.

The outputs of the receptors, bipolars, horizontals and amacrine cells in our “leaky integrator” model are their graded membrane potentials whereas the output of the ganglion cells is their “firing rate”. The firing rate of a cell depends only on the membrane potential of each cell

OPL Processing

IPL Processing

Overview of ganglion cell models

FIGURE 1.Top: overview of model structure. Cell types are: R, receptors; HC, horizontal cells; DBC and HBC, depolarizing and hyperpolarizing bipolar cells; PBD and PBH, positive part of bipolar cell potentials; ATD and ATH, transient amacrine cells from DBC and HBC channel; OPL, outer plexiform layer; IPL, inner plexiform layer. Bottom: ganglion cells RO-R4. The receptive field for ganglion cells type RO-R3 is composed of a small excitatory receptive field (ERF) and an overlapping larger inhibitory receptive field (IRF). The ERF and IRF in the R4 model are the same size. Input to both ERF and IRF are from bipolar and amacrine cells (pbd, phb, ath, atd). Spatial connections and other details of the algorithms are not shown here but are given in the text. [Reprinted with permission from Teeters and Arbib (1991).]

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JEFFREY

(each cell is modeled as a single compartment), follows the differential equation.

L. TEETERS

which

r, dm(t)/dt = -m(z) + s,(t). where m(t) denotes the membrane potential of that cell at time t; 5, is the time constant for the rate of change of this potential and S,(t) is the total input the cell receives from other cells. For ganglion cells, a threshold function r~ then converts m(t) into a firing rate M(t) = a@(t)). To specify the overall structure of our model, we must specify how each term S,(t) incorporates the input from all the other cells to which the given cell is connected. The notation B = W*A denotes a two-dimensional spatial convolution operation where the array of activity B is obtained by connecting array A to array B according to the connection weights in the mask W. Thus if layer m receives its input from layers of neurons whose firing rates are given by the arrays A and B, then the model represents the total input to m by a sum of the form W*A + W*B. Typically in this paper, a mask will be a center-on, surround-off type. 2.

METHODS

2.1. Stimulus representation While the single cone receptors in the retina have a density of about 5-30 cells per visual degree depending on the location (Carey, 1975), simulation tests have shown that a density of only 2 cells per visual degree allows sufficient accuracy for modeling responses to the stimuli we consider here. When the stimulus edge partially covers a receptor, we set the receptor inputs to values proportional to the area covered by the actual (analytical/continuous) stimulus. The error from the edge effect is then about 4% relative to the analytical solution (Teeters, 1989). We allow an arbitrary size and shape bitmap to represent our stimulus. In the simulations for the size and shape dependence of the ganglion cells, the velocity of the stimulation was set to 7.6 deg/sec so that the stimulus moves approx. 15 pixels in the grid each simulated second. 2.2. Computer simulation steps In computer simulation, we update the state of the network every Dt msec, proceeding through every cell type to compute new values of membrane potentials, and then forming new values of the firing rates in the case of ganglion cells. Step I. Updating the membrane potentials. The differential equation r,,, dm(t)/dt = -m(t) + S,,,(t) for the membrane potential is simulated ueiag the time step Dt to compute the new value m(t + Dt). Our simulation system allows one to write the model without reference to any numerical method, allowing the user to choose different numerical methods (e.g. the Euler method, trading off accuracy for speed) on different occasions without re-specifying the model. Step 2. Updating thejiring rates. For the ganglion cells, we convert m(r) into the firing rate M(t) = a(m(t)).

et al.

[a(x) = x if x > 0, 0 otherwise.] M(t) = m(t).

For

other

cells,

2.3. Forming the average response graphs To allow comparison between the model behavior and tabulated data, the temporal responses of the ganglion cells generated by the model are converted to an average response which is then scaled. The average response is calculated as the area under the response curve above threshold divided by the time from first to last response above threshold during the response to the leading edge of the stimulus (or, in other cases, from the beginning of the leading edge response to the end of the trailing edge response): ,gO TnRir!To where To is the time for the first such response and Tn is the time for the last. If the response decays in an exponential manner and is not actively abolished, the response duration will be infinitely long. For that reason the threshold used is not zero but a small positive number (0.001). The analogous experimental average is equal to the total number of spikes divided by the time from first to last spike during this period. Scaling is achieved by multiplying all calculated average responses by a “scaling constant’ so that the scaled average response to a 2 x 2 square moving at 7.6deg/sec matches that found experimentally by Ewert (1976). The scaling constant is given in the average response figure captions. 2.4. Numerical methods In all our simulations we chose to use the “Average Exact” (Teeters, 1989) method because it is stable under different simulation step sizes. This compares favorably with the Euler method that requires very small step sizes. The Average Exact method is derived on the basis of a heuristic which hypothesizes that the average value of the leaky integrator during a time interval is a more accurate basis for integration than the value at the end of the interval. 3. STIMULUS SHAPE AND SIZE DEPENDENCY In general, the average response of anuran ganglion cells to a moving stimulus depends on stimulus configuration, size, and velocity-a long thin bar moving in the direction of its long axis (a “worm” stimulus) will normally give a different response than the same sized stimulus moving perpendicular to its long axis (“antiworm”). Likewise, a square shaped stimulus will often generate a different response than do worm or antiworm stimuli. The response dependence on the edge length of moving worm, antiworm, and square-shaped stimuli has been determined in the toad (Ewert & Hock, 1972: Ewert. 1976) and in the frog (Schtirg-Pfeiffer & Ewert, 1981). The data sets are quite different even though the same anuran cell types are recorded. In the frog data, only the R3 cell shows a distinct difference in response

RETINA MODEL

to worm, antiworm, and square stimuli. Although T&A mainly tuned the ganglion cell models to frog data, this paper will use the toad data because toad ganglion cells show a much better ability to differentiate between stimulus types. Unfortunately, several things suggest that we should know more than most publications tell us about the way the experimentalists calculate average responses from the temporal spikes. The tabulated toad data imply that the R2 and R3 cell responses to worm stimuli are nearly independent of the edge length, which suggests that there is a response to only the leading or the trailing edge, but not both. However, later data from the same species (Tsai & Ewert, 1987) show that R2 and R3 cells respond to both the leading and the trailing edge of a worm stimulus. It turned out that Ewert (personal communication) only used a cell’s response to the leading edge of the stimulus to calculate the average response.* In accordance to this methodology, we relied on the leading edge response to calculate the average response-in all the cases the leading edge responses are clearly discernible from the residual responses. Our ability to match these data (and those analyzed by T&A) suggests that the model is indeed robust enough to serve as a valid “front end” for Rana computatrix (Arbib, 1987). A brief qualitative analysis of the model responses to various stimulus shapes and sizes could offer some useful guidelines for further tuning of the base model. An instantaneous response of a ganglion cell is the result of summation of ERF induced excitatory response and the IRF induced inhibitory response. The inputs to ERF and IRF could be of different combinations of channels (PBH, PBD, ATH, ATD) depending on the ganglion cell types. For instance, R2 receives PBH and ATH channels for its ERF, and ATH and ATD channels for its IRF. However, sustained bipolar channel (PBH, PBD) responses and transient amacrine channel (ATH, ATD) responses present different spatial characteristics. For example, the PBH bipolar channel layer forms an activation profile identical to the size and shape of the dark stimulus. The T&A model uses a high-pass filter to represent the amacrine cells as they convert sustained bipolar signals into transients. The resulting amacrine cell layer forms an exponentially decaying surface starting from the edges of the moving stimulus: the ATH layer forms such a surface starting from the leading edge of a dark moving surface, and the ATD layer from the trailing edge. If the shapes of the stimulus classes are restricted to rectangles and if each bipolar and amacrine cell has maximum instantaneous firing rate of 1, overall activities of PBH and ATH on their layers are: PBH,,, = I * h

s m

ATH,,,

=h .

e-xiV’Tdx

x=0

*Nevertheless,our temporal graphs (Figs 3, 5 and 7) show both the leading and the trailing edge.

2365

where x is the distance between the amacrines corresponding to the leading edge and the position of amacrines the stimulus has passed over. Obviously, PBH,, is a function of both stimulus length (1) and height (h) while ATH,, is only dependent on height of the stimulus for given velocity (u) and time constant (7). Thus, while the activation pattern on the PBH layer directly reflects stimulus shape and size, ATH layer activation pattern produces identical firing patterns for worm, antiworm and square so long as they have the same height. These different spatial firing patterns of bipolars and amacrines will form the basis of the shape dependence of ganglion cells. The average response of anuran ganglion cells usually increases with stimulus size smaller than the ERF. Simply, the increase in ganglion cell response in our model stems from the fact that, assuming response durations are about equal for a given velocity, as the stimulus size increases it excites a larger area of receptors and thus of bipolars and amacrines, so that it increases the instantaneous ganglion cell response which is proportional to the sum of activation of amacrines and/or bipolars within the ERF. However, bipolar and amacrine contributions to the response growth will be different in that bipolar channel contributions will increase proportional to the stimulus area but the amacrine channel contribution will increase proportional to the height. As the stimulus size increases beyond the ERF and into the IRF region, the IRFcontributed inhibition takes effect and reduces the total response. 3.1. Stimulus size dependence of R2 cells The T&A model failed to match the experimental data on stimulus size and shape dependency shown in Fig. 2(a) (Ewert & Hock, 1972), although there is some similarity in that the worm and antiworm responses peak at 4 deg edge lengths. Three changes were made to the model to enable matching the data: (i) Reduce ZRF strength : The R2 cells for which the T&A model were tuned follow the characteristic property of not responding to vertical bars longer than 10 deg (Griisser & Griisser-Cornehls, 1976). The data in Fig. 2(a) contradict this, because it reports that toad R2 cells respond to 16 deg vertical bars. This suggests that the inhibitory surround is weaker in these toad cells than in the R2 cells for which the T&A model was tuned. Because of this difference, the inhibitory receptive field sigmoid of the earlier model was reduced from 4.6 to 4 and the weighting from 3 to 2.3. (ii) Remove the ATD component: We achieved temporal firing profiles that are closer to the experimental data by removing the ATD component. (iii) Put the sustained bipolar component in the ZZW: The data show nearly identical responses for both antiworm and square of the same height. We saw earlier that the transient amacrine channel presents identical responses to stimuli so long as the height of the stimuli are identical, which is the case for the antiworm and

JEFFREY L. TEETERS et al.

2366

square used in the experiment. However, PBH input to the ERF would predict that its excitatory contribution for square stimuli will increase proportionally to the stimulus area (edge-size’) while that of antiworm will increase linearly as a function of the height (edge-size)predicting a higher increase for square than antiworm response as edge lengths are increased. To remedy this discrepancy, IRF is modeled as receiving input from the PBH channel as well. With this modification both instantaneous response and duration of response were reduced due to a larger effect of the square “straddling” between ERF and IRF, where the larger portion of square stimulus captured by IRF negating the increased excitation from ERF. It turns out that PBH input should be stronger to the ERF than to the IRF-the ratio (PBH

to ERF : PBH to IRF) that gives the best fit to the data is 0.3:O.l. The data also show that R2 cell responses to worms of different length are almost constant. In the model there exists (decreased) residual response for the long worms, while the body of the stimulus passes by the ERF after the leading edge, due to the sustained PBH input to the ERF. This residual response has a potential to reduce average response if the entire firing duration is considered, especially if the PBH contribution to ERF is low. With the parameter setting that satisfies all of the R2 characteristic responses, among them decreased sustained response to a moving stimulus that stops in ERF, the model shows slow decrease in average response as worm length increases beyond 4deg. In fact, Ewert

35 F 2 2

**square”

30 25 20-

g* lo15 -

5 -

class

R2

0 0

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EDGE SIZE

worm

antiworm

square

t

(b) 40 , 35 F

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_

“wormW .+.... _

3025 20 -

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lo15 5 01

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EDGE SIZE FIGURE 2. R2 response to square (S = 2, 4, 8, 16, 32 deg), antiworm (Y = 2, 4, 8, 16, 32 deg), and worm (H = 2, 4, 8, 16, 32 deg). (a) Experimental data. [Redrawn with permission from Ewert and Hock (1972).] (b) Response of our current tuned model.

The scaling

constant

used to calculate the average firing rate is 38.2. The ordinate abscissa is the stimulus size in degrees.

is the average

firing rate and the

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RETINA MODEL

: 2 g 4 2 E

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FIGURE 3. R2 model temporal responses of our current tuned model. The stimuli are: worm (W) 2, 4, 16, 32 deg; antiworm (AW) 4, 16, 32 deg; and square (S) 4, 16, 32 deg. Stimulus type and edge length are given with each plot. All graphs show firing rate versus time in seconds.

(1972; personal communications) and Garcia et al. (1988; personal communication, on frog R2) observed similar responses and Ewert only counted briskly firing leading edge response for his calculation of average response. Similarly, for the calculation of average response in our model the residual response due to sustained input from the PBH was not counted. With these changes it is possible to closely approximate the data, as we show in Fig. 2(b). Temporal responses are given in Fig. 3. It was interesting to find

that the model showed slightly greater average response to the square than to the antiworm when the edge lengths were < 4 deg but slightly lesser response when the edges were > 4 deg. This is what we observe in the experimental data and this behavior began to appear after the inclusion of PBH in the R2’s IRF. Note that because the R2 model receives input only from off detecting channels it will respond to the leading edge of a B/W contrast stimulus, and to the trailing edge of a W/B stimulus.

2368

JEFFREY L. TEETERS et al. T&A model for R2

Revised R2 model tc = kO*ATH - k 1*ATH !removing ATD PBH_e = 0.3 PBH, PBH-i = 0.1 . PBH R2 = g. (kO*PBH_e - kl*PBH-i + tc) !PBH to IRF with k0 = mask(4,2.4, 1) kl = mask(19.5,4.0,2.3)

tc = kO+ATH - k 1r(ATH + ATD) g = pos(tc) = 1 if tc > 0,O otherwise R2 = g. (kO*PBH + tc) with k0 = mask(4,2.4, l), kl = mask(19.5,4.6, 3)

3.2. Stimulus size dependence of R3 cells The experimental data is shown in Fig. 4(a). The T&A model does not match the data very well because the response to worms decreases with increasing edge length and there is a separation of the square and antiworm responses for large stimuli. Both of these effects are due to the inclusion of the slight trailing edge response generated by the model to long square and worm stimuli when calculating the average response. Only minor changes are needed to tune the model. Excluding the trailing edge (and relying only on the leading edge) response for the calculation of the average

responses (Ewert, personal communication) allows a good match to the data, although the response to the 32 deg square and antiworm is too large. Further, increasing the weight and the standard deviation of the IRF Gaussian mask and a little decrease in standard deviation of ERF mask allows a better match which is shown in Fig. 4(b). This suggests that the IRF receptive field is essentially like a plateau, with a very small decay with distance, while the ERF receptive field is like a sharp peak. Temporal responses for the tuned model, including the trailing edge response, are shown in Fig. 5. Comparison with the R2 temporal responses (Fig. 3) reveals that the main differences lie in (i) R2 responses show a

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EDGE SIZE FIGURE 4. R3 response to worm, antiworm, and square. (a) Experimental data. [Redrawn with permission from Eweti and Hock (1972).] (b) Response of our current tuned model, averaged over the leading edge response. Changes from T&A are an increase of the IRF strength, an increase in the standard deviation of the IRF Gaussian, and a reduction in the standard deviation of the ERF Gaussian. The scaling constant used to calculate the average firing rate is 44.0.

RETINA

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sustained component for the long worm and a sustained rebound for the large square stimuli, while (ii) R3 responses show transient responses for both the leading and the trailing edges of the long worm and large

square stimuli. The simulated temporal responses of R2 and R3 for the different stimuli approxjmate observed experimental data (Gaillard, personal communication) fairly well.

T&A model for R3

Revised R3 model

R3 = kO*a - (kl +a) defayed with a =p . ATD + ATH, where p = 0.4 and (s) delayed = signal s delayed 40 msec k0 = mask(8,2.4,1) kl = mask(19.5,4.6,

1.4)

and (s)

R3 = kO*a - (k 1*a) aWayed with a =p . ATD + ATH, where p = 0.4 delayed = signal s delayed 40 msec k0 = mask@, 2,1.15) kl = mask(19.5,10,2.38)

2370

JEFFREY L. TEETERS et ul. (a) I

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EDGE SIZE FIGURE 6. R4 response to worm, antiworm, and square. (a) Experimental data. [Redrawn with permission from Ewert and Hock (1972).] (b) Response of our current tuned model. The only change is a function which squashes the large magnitude responses. The tuned model response to a 32deg square goes below that of the antiworm because of extreme sensitivity of the “average response” to response duration (see text). The scaling constant used to calculate the average firing rate is 37.5.

3.3. Stimulus size dependence of R4 cells

The T&A model approximates the experimental data [Fig. 6(a)] qualitatively, but the quantitative response to large square and antiworm stimuli is too large. The model response to a 16 deg square is larger by a factor of 8 than the response to a 2 deg square as compared to the factor of 2.5 found experimentally. Attempts to reduce the response to the larger stimuli by increasing the strength and standard deviation of the Gaussian IRF led to the conclusion that it was impossible. This is because increases in the inhibition will cause a larger reduction in the response to stimuli with a short distance between leading and trailing edge than with a longer stimulus. For the case of a 2 and 16 deg square this can be explained as follows. The R4 cell operates by summing excitation generated at the off edge and inhibition generated at the on edge. The leading edge of a black 2 deg square generates a swath of excitation through the receptive field which is 2 deg wide. This is one-eighth of the 16deg diameter receptive field. A

I6 deg square will stimulate the entire receptive field and thus generate approximately 8 times as much excitation as a 2 deg square. This factor of 8 excitation advantage for the 16deg square cannot be reduced by inhibition because inhibition will reduce the response to the shorter stimulus more than it reduces the response to the longer stimulus. This is because a shorter stimulus will have both leading and trailing edges in the receptive field at the same time. The inhibition generated by the trailing edge will tend to cancel the excitation generated by the leading edge. This is not the case for the longer stimulus because the leading edge of the 16 deg square will pass entirely through the receptive field before any inhibition from the trailing edge is generated. Hence, excitation generated by the leading edge of a 16 deg square is not canceled by the trailing edge inhibition. A small standard deviation in the excitatory Gaussian can boost the small square response relative to that of the large square. However, this would effectively shrink the excitatory zone, creating a central ERF and surround IRF receptive field structure which is not found in R4 cells.

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RETINA MODEL T&A model for R4

Revised R4 model

R4 = kO*(ATH - x . ATD) withx=l

R4 = f(s) where s = max(kO*(ATH - ATD),O) f(s) = s/(s + 0.2) ! non linear squash k0 = mask(15.5,3.5, 1.0)

k0 = mask(15.5.3.5, 1.0)

The impossibility of matching the data in this way does not mean that the information flow used in the model is wrong. However, it does suggest that non-linear summation causes a saturation effect which tends to limit the responses to large stimuli. Therefore, the model is

tuned by adding a non-linear “squash” function to the ganglion cell. As shown in Fig. 6(b) this enables it to match the data in magnitude but causes the response of the 32deg square to fall below that of the antiworm. This is caused by extreme sensitivity of the “average

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FIGURE 7. R4 model temporal responses of our current tuned model. The tuned model has a slightly longer response to the 32 deg square (bottom row) causing its average response to become smaller than that of the antiworm.

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JEFFREY L. TEETERS et al.

response” metric to changes in the response duration. As mentioned previously, the average response is calculated as the area under the ganglion cell response curve divided by the response duration. The squash function causes the response of the 32 deg square to remain above the threshold longer, increasing the total time of response without appreciably increasing the area under the curve. A biological correlate to this problem is that a stray spike long after the main response can greatly alter the “first to last spike” average response measure. Further refinements of the threshold or the model would probably allow a closer match to the experimental data. However this would not lead to insight, because it is equivalent to start a “stray spike” long after the main response solely for the purpose of obtaining a desired average. Temporal responses of the “tuned” R4 model are shown in Fig. 7. 4. DISCUSSION

The essential features of the models presented in this paper that enabled a close match to the stimulus shape and size dependency data were also used by several earlier models which attempt to explain those properties in anurans. For example, the DOG center-surround structure was used to account for response in the R2 and R3 cells by an der Heiden and Roth (1987, 1989), Ewert and von Seelen (1974; also reported in Ewert, 1976) and by Griisser and Griisser-Cornehls (1973). Variations in the temporal filter characteristics of retinal elements have been used by Eckmiller (1975), Griisser (1967) and Grusser, Finkelstein and Griisser-Cornehls (1968) to account for variations in the velocity exponent. However, where the previous models were specialized to account for only particular phenomena, the models in this paper are not only able to account for the dependence on stimulus shape and size, but also able to account for the generation of characteristic ganglion cell response properties despite additional constraints applied to the ganglion cell models developed by T&A. For the R2, R3 and R4 cell models given in T&A, the response dependence on stimulus shape and size was tested in two parts. First, the original unmodified model was tested. Second, parameter adjustments and in some cases algorithm modifications, were made in an attempt to “tune” the model to attain a closer match to the experimental data on stimulus shape and size dependence. While the original untuned models did not quantitatively match the data, they were qualitatively correct. 4.1. Predictions based on the modzjied model behavior We now consider two important questions in detail: how do the changes made to the models here affect their

ability to account for characteristic addressed by T&A, and what predictions result from the changes? R2 cell: characteristic R2 responses as identified by Maturana, Lettvin, McCulloch and Pitts (1960), and Griisser and Griisser-Cornehls (1976) are a lack of response to a diffuse light change, a lack of response to

a moving antiworm longer than 10 deg, a prolonged response to a moving stimulus that stops in the ERF, a sensitivity to movement, a cessation of sustained response following a transient off of the general illumination, and a stronger response to small objects than to large objects. All of these characteristic responses but one are found in the toad R2 cells studied by Ewert and Hock-toad R2’s respond to antiworm stimuli up to 16 deg in length. The tuning performed in this paper does not destroy the ability of the R2 model to account for these properties. Specifically, reduction in the IRF weighting to the R2 model will not allow response to full field flashes because the total IRF weighting is still larger than that of the ERF. Sensitivity to movement is preserved because the modified R2 cell receives input from transient amacrine cells which respond only to change. However, inclusion of PBH input to IRF leads to: Prediction 1: for R2 cells tested by Ewert and Hock (1972), while the average response to squares and antiworms of the same height may be almost identical, the temporal responses (spike trains) will be d@erent as shown in Fig. 3.

R3 cell: Changes made to- the R3 model were the exclusion of the trailing edge response for the calculation of the average response, increase in the standard deviation of the Gaussian mask for IRF as well as in the IRF overall strength, and a little reduction in the standard deviation of the ERF Gaussian mask and a little increase in the weight. The tuned R3 model is still capable of generating all the characteristic responses of T&A. R4 cell: The only change made in the R4 cell model was the addition of a non-linear squash function which reduces the magnitude of response for favored stimuli. The non-linear squash function will not modify the characteristic property of the cell which is the response to off-going stimuli. 4.2. Future refinements of the retina model The shape/size tuned retinal model could be tested against other qualitative and quantitative data such as the average response as a function of velocity, adaptive state, etc. (Griisser & Griisser-Cornehls, 1976). In order to account for these data, we should probably incorporate more detailed physiological and morphological facts. Some of the most obvious ingredients could be:

(9 a more detail horizontal

cell model that is sensitive to the presentation of local stimuli; (ii) feedback loops among some layers (e.g. feedback from the amacrines to the bipolars); (iii) multi-compartmental dendritic processing and axonal transmission properties. Also we have to note that the modeling of transient amacrine cells was based on phenomenological observations rather than on detailed physiological data on these cells. It might be possible to express the comparatively more responsive synaptic transfer process of R3s by, for instance, decreasing the amacrine time constant.

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FIGURE 8. (a) The time-course of a stimulus’ up-side edge location induced by a simulated respiratory eye movement. We assumed that the eye oscillation of 0.5 deg/sec induced an equal amount of retinal image oscillation. (b) Schipperheyn’s data on respiratory eye movement: time-course of respiratory eye movements in unrestrained frog. The slight movements (frequency 1.3/set) have a maximum amplitude of 28 min arc. The slow movements at intervals of 15-20s~ are caused by periodic emptying of the lungs. (Note that the respiratory eye movement is the train of small sinusoidal curves. The slow rhythm is ignored because the response from the movement is negligible compared to that from the respiratory eye movement due to its much lower average speed.) Upward deviation of the optical axis is recorded downward. Time marking l/set. Temperature 25.6”C (Schipperheyn, 1963).

Teeters (1989) comments that the high-pass filter transient amacrine is unsuitable for the R4 cell model. This points out the need for an improved transient generating mechanism in the R4 cell. In retrospect, this is not surprising, because other properties of R4 cells, such as rhythmical bursting and delayed response to illumination decreases also cannot be accounted for easily by the high-pass filter mechanism used in the model. Rhythmical bursting also occurs in some R3 cells (Maturana et al., 1960), suggesting that a high-pass filter may also be inadequate to explain all of their response properties. Some type of negative feedback, with time delays or voltage dependent activation, is an obvious candidate mechanism which could generate oscillations in the neural potentials leading to bursting type response patterns. Further simulations will be needed to determine if such a mechanism can be made to simultaneously account for the characteristic R4 properties, the velocity exponent, rhythmic bursting, and the long response duration to a decrease in illumination. However, some of the characteristics R4 properties such as prolonged response to a stationary dark object and to the general illumination decrease can be achieved by incorporating the off channel bipolar inputs. In fact, Lee (1986) uses the sustained amacrine channel, PBH in our model, as the sole input to his R4 cells. It may also be possible to model the R4’s large time constant for a moving object with the proper formulation of spatially sensitive horizontal cells.

4.3. Providing a flexible framework retina

for modeling anuran

In summary, our current retina model cannot match all the experimental data, but does show how a relatively

simple model can explain a wide range of ganglion cell properties. It also makes clear how, by changing parameter values of different inputs to the ganglion cells, the response properties of the ganglion cells will in turn change. For instance, when the weight of the input from the PBH to the IRF of R2 cells is increased, the previously described average response of the cell will diminish as well as the strength of the sustained response. We should also note that retinal ganglion cells of the “same” type show a population of responses, as is elegantly shown in Gaillard’s (personal communication, 1991) experiment on R3 type cells. Gaillard’s result shows surprisingly large variances in ERF size, temporal activation patterns, etc., among the R3 cells. Similarly, we can expect that bipolars and amacrines will also form statistical distributions of responses. It may be that during embryogenesis a connection pattern from amacrines to a ganglion cell will be basically homogeneous, but that during postnatal development certain connections are strengthened while some are weakened thus giving the diversity among ganglion cells of the same type. The fact that reciprocal connections exist between the bipolar cells and amacrines gives some hope that a similar connectivity may exist between amacrines and ganglion cells, which could provide information paths for selective strengthening and weakening required for diversity. In our current model the amacrine population is represented by a layer of cells which share exactly the same properties. This has proven enough to match the experimental data described in this paper. But it is certain that the real retina contains several kinds of amacrine cells showing different properties, and this

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JEFFREY L. TEETERS et a/. (a)

ERR-size (5 cells/de@

ERF-size (S cells/de@ FIGURE 9. In the remaining figures, the density of cells in the simulation is 5 cells/deg. Thus five vertical bars along the horizontal axis amounts to 1 visual degree in Figs 9(akl I(b). (a) R3 response to 2 x 2 deg stimulus moving at 10 degisec. The R3 here is the “base” shape/sire tuned model-ERF = (8, 2, 1.15), IRF = (19.5, 10, 2.38). We see that the ERF size of the R3 is about 8 deg. (b) R3 response when the stimulus moves at 1 deglsec. All other parameter settings are identical to those used in Fig. 9(a). We can see that the effective ERF size is reduced to about 4 deg and that the peak excitation is also reduced to about one-quarter of that of Fig. 9(a).

could promote higher variability in the response profiles of the ganglion population whose response depend on amacrine input. For instance, in our preliminary studies on the velocity dependence of ganglion cells we found it beneficial to decrease the high-pass filter (amacrine) time constant from 300 to 50msec for the R3 and R4 ganglion cells to yield a better fit to the quantitative data. This suggests that the amacrine time constant may be better represented as forming a statistical distribution such as a normal dist~bution centered at a “typical” value and that the amacrines feeding into the R3 consists mostly of the values in the lower spectrum. The populational approach could also be applied to the ganglion cells. Thus, we are led to place more emphasis on the variation of response properties in a population of neurons of the “same” class, rather than questing for “the” neuron of a given type. One question that could arise when considering the popula~onal approach is whether there exists an illdefined boundary or just a “continuum” between different classes of ganglion cells. Should we construct a model so that it is possible for one category of cells to jump to another simply by, for instance, adjusting the “power” of a sustained input or the transient input? Gaillard (personal communication) has found “R3-like” units whose characteristic responses are similar to R3 units but whose velocity dependence is closer to that found in R2 ganglion &ells. Their response profiles are

stronger in intensity and amorally more extended than those of typical R3 units. R3s differ from R2s in that (i) their ERFs are larger, (ii) their ERFs receive no sustained input channel, and (iii) they have delayed IRFinhibition. We believe the significance of these differences increases in the order listed above. We also think the more important a characteristic is, the less flexible are the parameters that make the characteristic. Notice that the “discrimination curves” of R2 and R3 cells to different stimuli are su~~si~ly similar. The main difference lies in a shift of the optimal length of the square (S) and the antiworm (A) from 4 deg (R2 units) to 8 deg (R3 units) and consequently in a shift of the crossing point between Worm (W), S, and A curves. This difference can be accounted for by a simple difference in the R3’s ERF size and therefore we may predict that some R3 cells may have smaller ERFs so that their responses to dynamic visual stimuli are similar to R2 responses. 4.4. Respiratory R3 response Schipperheyn (1%3, 1965) claims that R3 responses can be elicited by a respiration induced, predominantly vertical, eye oscillatory movement of about 0.5 deg/sec. The existence of the respiratory R3 response is not widely accepted, even though such a response is possible in theory because the displacement and the velocity of the r~piration*indu~d retinal image shift are above

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RETINA MODEL

(a)

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FIGURE 10. (a) Here we changed the IRF weight from 2.38 to 1.15. Other parameters of ERF and IRF masks are identical to those used in Fig. 9(a)-ERF = (8, 2, 1.15), IRF = (1.95, 10, 1.15). The reduction of IRF-induced inhibition to overall R3 response is manifested in both an increase in peak excitation [15% increase compared to Fig. 9(a)] and in the ERF size (12.5 deg compared to 8 deg). Stimulus vel = 10 deg/sec. (b) When the stimulus velocity is reduced to 1 deg/sec, both the ERF size (about 4.2 deg) and the peak excitation level [one-quarter of Fig. 10(a)] are reduced. However, comparing Fig. 9(a, b) and Fig. lO(a, b), we can see that the reduction of IRF weight results in a more drastic decrease of ERF size as the velocity decreases. Whereas Fig. 9(b) showed a 50% decrease in the effective ERF size, this figure shows more than a 65% decrease.

the minima needed for R3 excitation and because the neuronal adaptation of R3s is incomplete.* Ingle (1976) found a retinal input in the pretectum of the frog, which he called R6, interspersed among the usual R3s. These cells with small RF sizes (3-6 deg) discharged for 10 set or more when a dark edge was moved into the field and was oscillated at 0.5 deg/sec, similarly to that produced by the respiratory eye movement. We hypothesize that Ingle’s R6 is the same as Schipperheyn’s respiratory R3, and so modeled these cells (Resp-R3) by using the R3 model with a resolution of 5 cells/deg-our base cell density of 2 cells/deg might not be enough to capture the details since that the oscillation amplitude is very small (0.5 deg). To stimulate its model retina, the oscillatory movement is simulated as follows. y(t) is the upper edge location of the oscillating stimulus. The oscillation can be described as repeating a movement that starts from an initial point, y(O), to the point “y(O) + amp”

*Garcia and Gaillard (1989) report that neuronal adaptation of a frog (Rana esculenra) is almost nil. tThe oscillation may also be formulated as y(r) = y(O) + amp/2 + amp/2 sin[(2n/period) . t - n/(2. period)]. This gives results similar to those from the one used in the model. SERF-size = velocity (deg/sec) . response-duration (set).

and back again to the starting seconds:

point

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~(t + dt) = y(t) + u(t) * dt + l/2 * a(t) * dt*, with a(t) =

Act when y(O) < y(t)
where acceleration magnitude Act = 2 * (amp/2)/ (period/4)*, time step dt = 40 msec, oscillation amplitude amp = 0.5 deg, and period is 1 sec.? Here the acceleration is kept constant until the stimulus reaches the “y(O) + amp/2” conversion point, the midpoint of oscillation, where the acceleration sign is reversed. We believe the simulated oscillation [Fig. 8(a)] is a good approximation of real movement [Fig. 8(b)]. Experimental measurement of ERF size of an anuran ganglion cell is calculated from single cell recordings by measuring a duration of the cell response to a stimulus moving at a given ve1ocity.S The size calculated depends on various characteristics of the stimulus, among them the velocity of the stimulus and the method used (Ewert, Krug & Schiinitz, 1979; Garcia & Gaillard, 1989; Gaillard, personal communication, 1991). Thus, the ERF and IRF masks used in the simulation should be regarded as reflecting an anatomy (dendritic field) of R3 receptive field that elicits a typical ERF size at a given reference stimulus velocity but that could also show

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ERF-size (5 cells/de@

ERF-size (5 cells/deg> FIGURE 11. (a) The RF parameters used here are the original “qualitative characteristic response tuned” masks of T&A-ERF = (8, 2.4, l.O), IRF = (19.5, 4.6, 1.4). Stimulus vel = IOdeg/sec. The dif&rences from the mask used in Fig. 9(a) are that the IRF mask here has a smaller standard deviation and a lesser weight and that the standard deviation of the ERF mask is a little increased. We can see that R3 exhibits an ERF size of about 5 deg. (b) Compared to Fig. 1l(a), when the stimulus velocity is reduced to 1 deg/sec, ERF size is reduced to about 3.2 deg. This reducton (35%) in effective ERF size is somewhat less than those experienced by R3s in Fig. 9(b) and Fig. 10(b). (a) 50

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FIGURE 12. (a) The shape/size tuned R3 response to a small (2 deg) dark edge oscillating at 0.5 deg/sec with the ATD component set to zero. frhe rest of the simulations on the oscillating edge are performed with the masks-ERF = (8,2, I. 15), IRF = (19.5, 10, 2.38).] The temporal Resp-R3 response is sampled at the conversion point. Minima occur at oscillation ends, “y(O)” and “y(O) + amp”, and the maxima occur at the conversion points during the advancing phases of the oscillation. (b) The effective ERF size of the R3 during the respiratory eye movement. The ERF size of the R3 varies between 1.4 and 4 deg during the oscillatory cycles. The average ERF size is 2.96 deg.

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RETINA MODEL

dynamic “effective ERF” sizes depending on stimulus characteristics such as shape, size, contrast, and velocity. In the simulation, we measured the effective ERF size of a cell from a spatial layer of cells, which gives identical results as those measured from temporal single cell recordings. Because an oscillating edge undergoes changes in its instantaneous velocity during the respiratory cycle, the response amplitude and the ERF characteristics of a R3 ganglion cell that depend on the stimulus velocity also show variation during the cycle. So, first we compared the responses of R3 to leading edge of a 2 x 2 deg dark object moving upon a bright background at constant velocity of 1 deg/sec, an average speed during the respiratory cycle, to those of moving at the reference velocity (10 deg/sec). With the earlier shape/size tuned parameters, the simulation revealed that the effective ERF of the R3 was about 8 deg at vel = 10 deg/sec and about 4.0 deg at vel = 1 deg/sec [Fig. 9(a, b), respectively]. The peak excitation of the stimulus at 1 deg/sec is also reduced to about one-quarter of that at the 10 deg/sec. The spatial profile of ERF and IRF masks also influences the R3’s ERF size as the velocity changes. If the IRF weight in the IRF mask is reduced, the resulting effective ERF size will be more dependent upon the length of the spatial amacrine firing pattern (length of exponentially decreasing tail) because the IRF signal, which in part is proportional to the spatial extent of the amacrine firing pattern within the IRF region, is

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reduced. When we decreased the weight of the IRF to that of the ERF, the effective ERF (12.5 deg) was reduced to about one-third (4.2 deg) when the velocity was decreased from 10 deg/sec [Fig. 10(a)] to 1 deg/sec [Fig. 10(b)]. The peak amplitude is also reduced to about one-quarter of that at the 10 deg/sec. The standard deviation of the masks influences the ERF size as well. When tested with T&A’s base parameters [ERF =(8, 2.4, l), IRF = (20, 4.6, 1.4)], the effective ERF size at vel = 1 deg/sec [3.2 deg; Fig. 1l(b)] is about 65% of that at the 10 deg/sec [5 deg; Fig. 1l(a)]. We observed that it is the general tendency of ERF size to decrease with lower velocity. This becomes more prominent if the ERF mask’s standard deviation is decreased. This is due to the fact that if the ERF mask has a sharp peak, then as the spatial amacrine firing profile shrinks with a velocity decrease, R3’s ERF size is more affected by the shrinkage because the mask gathers less input from its peripheral regions when the mask is being convolved off-center to the amacrine firing pattern. From the above constant velocity simulations we have seen that an R3 could respond to a small object moving at respiratory speed, albeit with lesser intensity and smaller ERF size. The basic shape/size tuned R3 is then tested on the upper edge of the simulated oscillating stimulus of a 10 x 2 deg dark object without taking the cell’s neuronal adaptation into account. (The temporal response of the R3 neuron was sampled from the cell

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FIGURE 13. (a) The RespR3 response when the ATD channel contribution is set to 0.35. Due to the ATD contribution, RespR3 is made a little sensitive to the retreating edge. We can se-e the effect of this through the low/smooth humps around the conversion points during the retreating phases of the edge. (b) The effective ERF size of the R3 during the respiratory eye movement with the ATD component set to 0.35. The ERF size of the R3 varies between 3.2 and 4.2 deg and the average ERF size is 3.87 deg.

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Time (set) FIGURE 14. (a) The Resp-R3 response when the ATD channel contribution is set to 1. Two peaks of activation for one cycle of stimulus oscillation is observable. Compared to Fig. 12(a), the most prominent change lies in the increase (103%) of the average Resp-R3 response level. (b) The effective ERF size of the R3 during the respiratory eye movement with the ATD component set to 1.0. The ERF size varies between 3.8 and 4.6deg and the average size is 4.29deg.

that is located at the conversion point.) With ATD channel contribution set to zero, the peak excitation occurs when the edge passes the conversion point, where the velocity is highest, and continues to decrease until the initiation of a new oscillation period where it repeats an identical oscillatory response [Fig. 12(a)]. The peak excitation is about one-fifth of that when the stimulus moves at the continuous velocity of 1OdegJsec [Fig. 9(a)]. Effective ERF size also varied during the oscillation cycles [Fig. 12(b)]. It was largest (4 deg) at the conversion points in the advancing phases and smallest (1.4 deg) at oscillatory ends. The average ERF size was 2.96 deg. When the ATD channel contribution is set to 0.35, the R3 becomes a little sensitive during the retreating phase of the edge. As expected, R3 response shows low/smooth humps around the conversion points during the retreating phases of the edge [Fig. 13(a)]. The effective ERF size is also affected by the input from the ATD channel [Fig. 13(b)]. By setting the ATD channel cont~bution to 1 and thus making the R3 fully sensitive to the advancing brightness during the retreating phase of the dark edge, R3 response shows two cycles of oscillation during one period of respiratory oscillation [Fig. 14(a)]. The one peak occurs when the edge passes the conversion point during the advancing phase (contribution of ATH channel) and the other happens when it passes the point during the retreating phase (contribution of ATD channel). The effective ERF size varies between 3.8 and 46deg and

the average size is 4.29 deg [Fig. 14(b)]. Compared to the response when the ATD channel was zero [Fig. 12(a)], the peak amplitude is a little increased (24%) but the average intensity is doubled. In conclusion, as observed in the simulations, the respiratory R3 response can occur and its ERF size is significantly (2~O~) smaller than normal R3s responding to faster moving objects. And further, specific tuning such as the reduction of IRF inhibition made R3s more sensitive to the respiratory movement and fit the ERF size of Ingle’s R6 when an edge was oscillated at 0.5 deg/sec. Such findings support the hypothesis that Ingle’s R6 may in fact be one subclass among the spectrum of R3s. Schipperheyn (1963) and Griisser and GriisserCornehls (1976) suggested that respiratory R3 cells may play an important role in the detection of stationary objects. Even though the peak response to the oscillating edge is small, the respiratory R3 response could furnish excellent information on the boundaries of stationary objects because of its small ERF size and the continuous response. We have also seen that by increasing the ATD channel contribution, the respiratory R3 response is made stronger and steadier. It is interesting to see that the R3 response of Ifyia, one of the anurans with the highest sensitivity to stationary objects, appears to rely more on the trailing edge response (ATD contribution) than other anurans like Rana and Bufo (Griisser-Cornehls, 1988).

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RETINA MODEL

When simulating the Resp-R3 responses, we ignored one important aspect of ganglion cell characteristics. Even though neuronai adaptation is not complete in the R3s, oscillatory movement is the perfect stimulus movement that can cause the adaptation. Prediction 2: Cells that may be maximally sensitive to Resp-R3 movement should be the ones that show least neuronal adaptation. Also, cells that show relatively small ERF size, low IRF inhibition and higher response to the stimulus moving at higher speeds could be more responsive to the respiratory induced movement of stimulus. REFERENCES Arbib, M. A. (1987). Levels of modeling of mechanisms of visually guided behavior. 3e~viorat and Brain Sciences, 10, 407-465. Carey, R. G. (1975). A quantitative analysis of the distriburion of the retinal elements in frogs and toads with special emphasis on le Area Retinalis. Masters thesis, University of Massachusetts at Amherst,

Amherst, Mass. Eckmiller, R. (1975). Electronic simulation of the vertebrate retina. IEEE Transactions

on Biomedical

engineering,

BME-22,

305-3 11.

Ewert, J.-P. (1976). The visual system of the toad: Behavioral and physiological studies on a pattern recognition system. In Fite, K. V. (Ed.), The amphibian visual system (pp. 142-202). New York: Academic Press. Ewert, J.-P. & Hock, F. (1972). Movement-~nsitive neurons in the toad’s retina. Experimental Brain Research, 16, 41-59. Ewert, J.-P. & Seelen, W. von (1974). Neurobiologie und SystemTheorie eines visuellen Muster-Erkennungsmechanismus bei K&en. Kybernetik, 14, 163-183. Ewert, J.-P., Krug, H. & Schijnitz, G. (1979). Activity of retinal class R3 ganglion cells in the toad Bufo bufo (L.) in response to moving configurational stimuli: Influence of the movement direction. Journal of Comparative

Physiology,

129, 21 l-215.

Gaillard, F. & Garcia, R. (1986). The velocity function of ipsilatateral visual units in the frog optic tectum: Comparison with retinal ganglion cells. Neurosci~~e Letters, 65, 99-103. Garcia, R. & Gaillard, F. (1989). Quantitative studies on ipsilateral type 2 retinotectotectal (IRTT) units in frogs: Homologies with R3 ganglion cells. Journal of Comparative Physiology A, 164, 377-389.

Garcia, R., Gaillard, F. 8~ Jacquenod, J.-C. (1988). A quantitative analysis of ipsilateral tectal unit response to moving stimuli to moving stimuli in frogs. Journal of Compararive Physiology A, 162, 443-451.

an der Heiden, U. & Roth, G. (1989). Retina and optic tectum in amphibians: A mathematical model and simulation studies. In Ewert, J.-P. & Arbib, M. A. (Eds), Visumotor coordination: Amphibians, comparisons, models, and robots (‘pp.243-267). New York: Plenum. Ingle, D. (1976). Behavioral correlates of central visual functions in anurans. In LlinBs, R. & Precht, W. (Eds), Frog neurobiology (pp. 435-451). New York: Springer. Lee, Y. B. (1986). A neural network model of frog retina: A discrete rime-space approach. Ph.D. dis~rtation, Depa~ment of Computer and Information Science, University of Massachusetts at Amherst, Mass. Maturana, H. R., Lettvin, J. Y., McCulloch, W. S. & Pitts, W. H. (1960). Anatomy and physiology of vision in the frog (Ranapipiens). Journal of General Physiology

(SuppI.], 43, 129-175.

~hip~rheyn, J. J. (1963). Respiratory eye mov~ent and perceptions of stationary objects in the frog. Acta Physiologica et Pharmacotogica Neerlandica, 12, 157-l 59. Schipperheyn, J. J. (1965). Contrast detection in frog’s retina. Acra Physiologica

et Pharmacologica

Neertdandica,

13, 231-277.

Schiirg-Pfeiffer, E. Br Ewert, J.-P. (1981). Inv~tjgation of neurons involved in the analysis of gestalt prey features in the frog Rana temporaria. Journal of Comparative Physiology, 141, 139-152. Teeters, J. L. (1989). A simulation system for neural networks and model for the anuran retina. Technical Report 89-01, Center for Neural Engineering, University of Southern California, Los Angeles, Calif. Teeters, J. L. & Arbib, M. A. (1991). A model of anuran retina relating interneurons to ganglion cell responses. Biological Cybernetics, 64, 197-207.

Tsai, H. J. & Ewert, J.-P. (1987). Edge preference of retinal and tectal neurons in common toads (Eufo bufo) in response to worm-like moving stripes: The question of ~haviorally relevant “position indicators”. Journal of Comparative Physiology A, 161, 295-304. Weitzenfeld, A. (1990). NSL, neural simulation language, version 2.0. Technical Report 90-01, Center for Neural Engineering, University of Southern California, Los Angeles, Calif.

Acknowledgements-The research reported in this paper was supported in part by NIH under grant number ROl NS24926 from NINDS. The preliminary form of many of the results reported here were obtained with computer time provided by the San Diego Supercomputer Center and may be found in the thesis by Teeters (1989). All results in this paper were obtained from programs subsequently written in the object-oriented Neuron Simulation Language NSL 2.0 (Weitzenfeld, 1990) which is an extension of C+ + running on SUN 4 workstations. We express our deep appreciation of F&d&c Gaillard for his comments on an earlier draft of this paper, and for showing us data which emphasize the diversity of response of ganglion cells of a given “type”.

Griisser, O.-J. (1967). Ein Analogmodell der Funktion bewegungsempffndlicher Neurone der Froschnetzhaut. P@?geersArchiv, 294, 65.

Griisser, O.-J. & Griisser-Cornehls, U. (1973). Neural mechanisms of visual movement perception and some psychophysical and behavioral correlations. In Jung, R. (Ed.), Handbook of sensory physiology (Vol. VI1/3A, pp. 333-426). New York: Springer. Griisser, O.-J. & Griisser-Cornehls, U. (t976). Neurophysiology of the anuran visual system. In LlinBs, R. & Precht, W. (Ed@, Frog neurobiology (pp. 297-385). New York: Springer. Griisser, O.-J., Finkelstein, D. & Gtisser-Cornehls, U. (1968). The effect of stimulus velocity on the response of movement sensitive neurons of the frog’s retina. Pfliiers Archiv, 300, 4966. Gtisser-Comehls, U. (1988). Neurophysiolo~~l properties of the retinal ganglion cell classes of the Cuban treefrog, Hyta septenrrionalis. Experimental

Brain Research,

73, 39-52.

an der Heiden, U. & Roth, G. (1987). Mathematical model and simulation of retina and tectum opticum of lower vertebrates, Acta Biorheorerica,

36, 179-2 12.

APPENDIX List of Abbreviations

NSL T&A R H (HC) RO-R4 HBC DBC PBH PBD ATH ATD ERF IRF

Neuron simulation language Teeters and Arbib model of the anuran retina (1991) Receptor cell Horizontal cell Retinal ganglion cell types: classes O-4 Hyperpolarizing bipolar cell Depolarizing bipolar cell Positive component of the h~rola~~ng bipolar cell Positive component of the depolarizing bipolar cell Hyperpolarizing transient amacrine Depolarizing transient amacrine Excitatory receptive field Inhibitory receptive field