Direction-selective simple cells in cat striate cortex: a developmental model

Direction-selective simple cells in cat striate cortex: A developmental model

Basabi Bhaumik and Akhil R.Garg

Department of Electrical Engineering, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi-! 10016, India.

Email: bhaumik@ee.iitd.ac.tn, akhil_garg@ieee.org

Abstract - A number of models have dealt with the receptive field (RF) properties of direction selective (DS) simple cells, but none address the question how do these RFs develop. Here we present a model for development of RFs for direction selective cortical cells based on the mechanism of diffusive cooperation and resource limited competition among the afferents converging on a cortical cell. Mean direction selective index (DSI) obtained 032179 matches closely with experimental data.

1. INTRODUCTION

Hubel and Wiesel [1], in their pioneering study of simple cells in the cat's visual cortex, discovered one of the most prominent properties of simple cells: their direction selectivity. Understanding the microcircuit transforming center surround receptive field of neurons in LGN into simple cells receptive field showing orientation and direction selectivity, a characteristics of cortical area 17 has been the aim of lasting research by anatomist, physiologist and modelers. The first models to be thoroughly evaluated theoretically and experimentally came from information theory, simple cells as Gabor filters tuned to stimulus orientation and spatial frequency [2][3][4] and from psychophysics, direction selectivity as a non-linear correlation process [5] or by spatio-temporally oriented linear Filters [6] [7]. Other theoretical and neurophysiological studies pointed out that the origin of direction selectivity could be related to the linear space-time receptive field structure in which response timing changes gradually across the field [8][9][10][ll]. The discovery of non-lagged and lagged cells [12][13][14] in the LGN of cats has given support to this hypothesis. Non-lagged and lagged cells that project to the cortex and are combined linearly could provide the temporal offset that is necessary to create direction selective receptive field. Experimental support for this hypothesis comes from several sources. In extra cellular measurements an analysis of the time structure of cortical responses shows mixture of lagged and non-lagged like timing in the response of direction selective cells [14][15].

In intracellular recordings of direction selective cells [15] [16] also demonstrated two underlying temporal components that match the temporal responses of lagged and non-lagged inputs respectively. Wimbauer et al. [15]

proposed the first developmental model based on this hypothesis. The degree of direction selectivity obtained in this model depends critically on the amount of correlation between lagged and non-lagged inputs. If the development is done with strong correlation the resulting spatiotemporal receptive fields are separable and direction selectivity of

cells is weak. On the other hand if the development is done with weak correlation the model yields good direction selectivity but the two receptive field formed from lagged and non-lagged LGN cells have different orientation specificity.

This implies that in direction selective cells orientation preference of the cell changes with time. There is little experimental support for receptive fields with preferred orientation that drifts in time [17][18][19],

We present a model based on the mechanism of diffusive cooperation and resource limited competition among geniculate afferents converging on a cortical cell. In our model the cells with high degree of direction selectivity have no drift in orientation preference with time. The receptive fields formed for lagged and non-lagged LGN cells have similar orientation preference but are displaced in space.

Our model consists of three layers retina, LGN and cortex. For modeling retina and LGN we have extended and modified the detailed model of Worgotter and Koch [20]. We have applied drifting sinusoidal grating to characterize the cortical cell in our simulated cortex. We get cells representing all types of preferred orientation and direction selectivity index (DSI). The DSI obtained matches the experimental results.

II. DEVELOPMENT OF CONNECTIVITY BETWEEN LGN AND CORTEX

A. Elements of the model

The basic assumptions of the model are (a) Competition for a presynaptic resource where a presynaptic cell has a fixed amount of resource to distribute among its branches. This constrains the number of axonal branches a neuron can maintain; (b) Competition between axons for target space. The axons are competing for neurotrophic factors released by the postsynaptic cells upon which axons innervate and (c) Diffusive cooperation between near neighbor (i) Cortical cells (ii) Same type of LGN cells. These biologically plausible assumptions were used in [21] to obtain orientation selectivity of cortical cell. We extended Bhaumik and Mathur's [21]

model for RF development in orientation selective cells to RF development in direction selective cells. Incorporating, competition between lagged and non-lagged LGN cells for target space in the cortex.

For the development of spatial receptive field we considered

two layers, LGN and cortex. The LGN is composed of four

MxM arrays, two arrays for non-lagged ON/OFF center cells

and other two arrays for lagged ON/OFF center cells. The

LGN cells project topographically to the cortex, with cell

arborizing over a preferred patch of cortex. The cortex is taken to be an NxN array with each cell supporting a dendrite area of 13x13. While axons are allowed to sprout and retract, the positions of dendrites are fixed relative to the cortical array. Non-lagged ON/OFF synapse represents the synaptic strength from a non-lagged ON/OFF center LGN cell to a cortical cell respectively. While lagged ON/OFF synapse represent the synaptic strength from lagged ON/OFF center LGN cell to cortical cell respectively. Time evolution of synaptic strength represents cortical development. Initially, from a given LGN location a model cortical cell receive synapses of nearly equal strength from lagged ON/OFF and non-lagged ON/OFF LGN cells lying one over the other in four sheets of neurons. How many axonal branches an LGN cell can support is determined by competition for presynaptic resource where a presynaptic cell has a fixed amount of resources to distribute among its branches. From any given location a cortical cell receives only one type of input, either from ON or OFF LGN cell.

Along with competition between ON and OFF LGN cells, there is cooperation among neighboring cells of same type i.e. ON(OFF) type non-lagged cells cooperate with neighbouring ON(OFF) type non-lagged cells. Similar cooperation takes place among neighbouring lagged cells of same type. Large ON (OFF) Synapse helps neighboring ON (OFF) synapses to grow and force OFF (ON) synapses out of neighborhood. This leads to the formation of sub fields within a cells receptive field. During synaptic growth, at a given time we assume that either non-lagged or lagged type of LGN cells are active as the lagged cell has delayed response. At the end of development we have two receptive field of-a single cortical cell, one a connectivity pattern from lagged LGN cell and the other connectivity pattern from non-lagged LGN cells. We also have cooperation between neighboring cortical cells. This ensures smooth variation in orientation selectivity in the simulated cortex.

NxN

) sum of the square of synaptic

B. Methods

In the model Wu represents the synaptic strength of the connection from the non-lagged ON-center LGN eel! at position J in LGN layer to the cortical cell at position I in the cortical layer. A cortical cell receives axons from 13x13 region of the LGN cell layers, centered at its retinotopic position in the LGN. The synaptic strengths

updated using the following update rule.

w,r

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= \\ (W£}

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strength of all branches emanating from a LGN cell at the location J. YJ represent fixed pre-synaptic resources available in the LGN cell at location J. The term (Yrki) enforces the competition among LGN cell axons and constrains the number of axonal branches the LGN cell can maintain.

where V.\ represent the sum of the square of synaptic strength of all four branches of LGN cell converging on the cortical cell at location I. y2 represent fixed post-synaptic resources available in the cortical cell at location I. (72-^2) enforces the competition among LGN cells for taget space in the cortex. AR

is the arbor function [22](Miller, 1994). DL and Dc are the diffusion constants in the LGN and cortex respectively.

Equation for updating Wu~ , W , j+, W , j ~ can be written similarly.

III. ANALYSIS OF RFs USING THREE LAYER VISUAL PATHWAY MODEL

A feed-forward three-layer visual pathway model shown in figure 1 is used for analyzing the response properties of cortical cell. Retinal layer is modeled as two 2D sheets lying one over the other, first containing 30x30 ON center cells and the other containing 30x30 OFF center cells. Spatial RFs of retinal cells for computational convenience are separated into two types center and surround, which we obtained using gaussian functions. The parameters in gaussian functions were adjusted to give RF similar to those found experimentally. The input stimulus is convolved with center and surround spatial RFs. The convolution results thus obtained are further

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Figure 1: Visual pathway, Retinal cells further divided into ON, OFF center cells giving output to ON center (Laggcd/Non-Laggcd) and OFF center (Lagged/Non-Lagged) LGN cells respectively.

LGN cells terminate in area 17 of cat Visual Cortex forming simple cells receptive field

convolved with temporal impulse response functions of center and surround, giving center and surround responses.Resultant analog response of ON-center cell is calculated by subtracting surround response from center response. Similarly, resultant analog response of OFF center cell is calculated by subtracting center response from surround response. Analog responses so obtained are then converted into spikes. For the details of retinal cells spatial receptive field, temporal response functions and mechanism of generation of spikes see the model used in [20] [23].

LGN layer was modeled as four 2D sheets lying one over the other, each containing 30X30 cells. Each sheet has separate type of cells i.e. non-lagged ON center, non-lagged OFF center, lagged ON center and lagged OFF center cells respectively. The lagged non-lagged distinction in LGN cells is not one related to spatial domain as both mimic the spatial properties of their retinal afferents. The two-cell group differs substantially in their response timing. The impulse response characteristics of the two types (Lagged/Non- Lagged) cells are as shown in figure 2. The functional form that we have used for temporal impulse responses of lagged and non-lagged LGN cells was used previously by Wimbauer et al (1997)[15] based on theoretical consideration due to Dong and Atick (1995)[24].

Figure 2. Impulse response characteristics for non-lagged and lagged LGN cell respectively.

Each ON center retinal cell provides afferent input to ON center lagged and non-lagged type of LGN cells and each OFF center retinal cell provides input to OFF center lagged and non-lagged LGN cells, lying at the same retinotopic position. The processing done by LGN cell is only in temporal domain. The mechanism for generating LGN spike is as follows: the spike train of retinal cell is convolved with corresponding impulse response function of LGN cells, to this is added a refractory voltage which depends upon the last firing time of the same LGN cell. If the total so obtained is more than the threshold the LGN cell produces a spike.

Following this process the spike train for each LGN cell is obtained. The response of non-lagged ON type LGN cell,

R LQN is expressed as

(2a) Where R, ONR G Ci s the spike output of ON center retinal cell, LN(t) is the impulse response function of non-

lagged LGN cell, vref(t-tf) the value of refractory voltage which depends on the last firing time of the same cell. ® Represents convolution. Output thus calculated is converted into spike train using the expression.

LGN

1

0 otherwise

The spike responses for non-lagged OFF LGN cells are calculated by using expression.

(2b)

LGN

1

0 otherwise

The spike response for lagged ON and OFF LGN cells are obtained by replacing LN (t) with LL(t) in equations 2(a) and 2(b). LL(t) is the impulse response function of lagged LGN cell. Each cortical cell is receiving direct afferent inputs from LGN cells; they have adjacent excitatory and inhibitory sub fields having projections from ON (non-lagged/lagged) center and OFF (non-lagged/lagged) center LGN cells respectively.

We have modeled 50x50 cortex where each cortical cell receive input from 13x13 LGN cells of all four types from the same retinotopic location. Prior to obtaining the response of cortical cell, the RFs are formed using the developmental model outlined in section II. Now the response of cortical cell at a given time is calculated using SRM (Spike response model)[25], the equation for calculating the response is,

k=1 J tfl^t,

Where, Ui(t) is the membrane potential of cortical cell at location I at time t, w u is the synaptic strength of LGN cell at location J with cortical cell at location I, of type k. e(t-t(L) is the EPSP (excitatory post synaptic potential) contributed by LGN cell spiking at time to,, t| is the last spiking time of LGN cell at the J location, t2 is the earliest spiking time between t]

and t]-90 msec. T|(t-tfc) is the refractory voltage of cortical cell, tfc is the last spiking time of cortical cell, Rp is the rest potential of the cortical cell, p is the synaptic scaling factor in [26]. Note that we have slightly modified SRM model by incorporating p. The output so obtained is 1 if the value of U[(t) is more than threshold, otherwise it is zero. We have used - 7 0 mv as resting potential and -35 mv as threshold potential for simulation result presented in the paper. Following the convention used by physiologists the total no. of spikes/sec/trial for each cortical cell is calculated. The degree of directional selectivity is then calculated, which depends upon the relative response strengths to motion in preferred

(RPD) and non-preferred direction (RNPD), it can be quantified as follows.

DSI= (RPD-RNPD)/(RPD+RPND)

IV SIMULATION RESULTS

Since the model requires connectivity pattern from LGN to cortex, first the weight updating was simulated on a 440 MHz Sun Ultra 10 m/c with 512 MB RAM for a 50x50 cortical layer and four overlapping 30x30 LGN cell layers using circular boundary conditions.

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Receptive fields in a 10x10 section from simulated cortex are as shown in figure 3. The red and green colors denote excitatory and inhibitory sub-region respectively. Out of the two 10x10 patches one is for input from non-lagged and other from lagged LGN cells. Each cortical cell had a 13x13 receptive field from non-lagged LGN cells and 13x13 receptive field from lagged LGN cells. Initial weights of the order of 10"6 were picked up from a uniform random distribution. The differential equation for weight updating was simulated in difference mode using synchronous weight update. At around 500 iterations a cortical cell's receptive field from both lagged and non-lagged type of cell can be seen to have small patches, each patch being mainly contributed by either ON or OFF LGN cells. Because of diffusion in the LGN whenever there are a majority of ON (OFF) synapses in a cortical cell's receptive field, slowly over time a clustering effect takes place and these ON (OFF) synapses help other neighboring ON (OFF) synapses to grow and push off any OFF (ON) synapses present in the patch. The same process is there for both lagged and non-lagged type of cells. As a result at around 2000 iterations well-formed receptive fields with one, two or three sub-fields are observed as shown in figure 3.

270

(b)

Figure 3 Receptive Fields of a 10x10 section of simulated cortex, (a) One from non-lagged type LGN cells and (b) other from lagged type LGN cells.

Red color indicates synapses from ON center LGN cells and green color indicates synapses from OFF center LGN cells. Intensity of color denotes the strength of synapse.

Oricntatloru deg)

Figure 4. Showing Polar Plot, Receptive Field and Response Plot of a sample cell from simulated cortex, which is having high DSI.

The moving sinusoidal gratings of all orientations (from 0 to 360 in step of 18 ) at 0.5 cycles/deg and temporal frequency of 2 cycles/sec was shown as input. Orientation from 0°to 180°

was assumed to be one direction and from 180° to 360° the other direction. Figure 4 shows polar plot, receptive field and response plot of a sample cell from simulated cortex. As seen from the polar and response plot the cell is direction, this is because the receptive field corresponding to lagged and non- lagged type of LGN cells are displaced in space and both are having similar orientation in space. When a sinusoidal grating is moving in one direction the contribution of lagged LGN cells does not add with contribution from non-lagged LGN cells, whereas for other direction the delayed contribution

synchronizes with the contribution from non-lagged LGN cells and therefore high response is obtained.

21 270

Figure 5. Showing Polar Plot, Receptive Field and Response Plot of a sample cell from simulated cortex, which is having low DSI but is orientation selective.

Similarly for another sample cell the polar plot, receptive field and response plot are shown in figure 5, as seen from the polar plot and the response plot the cell is not direction selective but orientation selective. Here the non-lagged LGN cells dominate the response of the cell. The contribution of lagged LGN cells however makes the response in one direction higher than the other as can be seen in the response plot.

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Figure 6 Showing percentage distributions of cells for different DSI maroon coior our data, yellow color dala adapted from DeAngelts el al.[6] blue color data adapted from Wimbaucr et al. [24]

For all the 2500 cortical cell the Direction selectivity index (DSI) was calculated, max DSI obtained was 0.8382 and smallest 0.0. The mean DSI obtained was 0.32179, which is very near to the value reported by DeAngelis et al. [17] as 0.26 in kittens. The cell with DSI more than 0.5 was termed as direction selective and cells with DSI more than 0.3 but less than 0.5 was termed as direction biased and the rest were termed as non-direction selective cells. As shown in

figure 6 there are 50% cells which are either direction selective or direction biased. The Wimbauer et al. [15] report lesser number of such cells even for the case when the development is done without any correlation in lagged and non-lagged type of LGN cells. Percentage of direction selective cells in our simulation result matches with reported [17] experimental data. In our simulated cortex we get all types of cortical cells that are direction selective, direction biased and non-direction selective distributed over the simulated cortex. From the simulated cortex we have picked six cells as shown in figure 7. The first four cells are direction selective with different preferred orientation. The fifth cell is direction selective but poorly tuned and the last cell is neither direction selective nor orientation selective.

Figure 7. Polar plots of sample cells out of simulated cortex. The first four cells are direction selective with preferred orientation of 0,90,180 and 270 degrees. The fifth cell is direction selective but has poor orientation selectivity. The last one is a non-tuned, non-direction selective cell

V DISCUSSION

According to Hubel and Wiesel selectivity of simple cells are

exclusively created by the pattern of convergence of inputs

from geniculate afferents onto simple cells. They suggested

that when individual afferents to a cortical cell have different

temporal response pattern the cortical cell would become

direction selective. This hypothesis got experimental support

with the discovery of lagged LGN cells [12][13]. In this paper

we have presented a model for receptive fieid formation in

direction selective cells based on biological plausible

competition and cooperation for growth and retraction of

axons in the geniculocortical pathway. We have studied the

response of cells in our simulated cortex using a three layer

feed forward model by showing drifting sinusoidal grating as

input stimulus. The model verifies that if there are cells in

LGN with different temporal response properties, and then

cells in cortex receiving input from such cells can show

direction sensing properties.

V REFERENCES

1. Hubel, D.H. & Wiesel, T.N. (1959). Receptive fields of single neurons in the cat's striate cortex, J, Physiol. 148, 574-591.

2. Daugman, J.G. (1985) Uncertainity relation for resolution and space, spatial frequency, and orientation optimized by two- dimensional visual cortical filter. JOSA A 2, 1160-1169.

3. Jones, J.P. and Palmer, L.A. (1987) The two-dimensional spatial structure of simple receptive fields in cat striate cortex.

J.Neurophysiol. 58, 1187-1211.

4. Marceleja, S. (1980) Mathematical description of the responses of simple cortical cell. JOSA 70,1297-1300.

5. van Santcn J. & Sperling G. (1984) A temporal covariance model of human motion perception JOSA 1,451.

6. Adetson, E.H. & Bergen, J.R. (1985) Spatiotemporal energy models for the perception of motion. JOSA A 2,284-299.

7. Watson, A.B. & Ahumada, A.J. (1985) Model of human visual- motion sensing. JOSA A 2, 322-342.

8. Albrccht, D.G. & Geisler, W.S. 1991. Motion selectivity and the contrast-response function of simple cells in the visual cortcs.

Visual Neurosci. 7,531-546.

9. McLean, J, Raab, S. and L.A.Palmcr. (1994) Contributions of linear mechanisms to the specification of local motion by simple cells in areas 17 and 18 of the cat. Visual Neuroscicnce, 11:271- 294.

10. Movshon, J.A., Thompson, LD. & Tolhurst, D.J. (1978) Spatial summation in the receptive fields of simple cells in the cat's striate cortex. J. Physiol. 283, 53-77.

11. Reid, R.C., Soodak, R.E. and Shapley, R.M. (1991) Direction selectivity and spatiolemporal structure of receptive fields of simple cells in cat striate cortex. J. Neurophysiol. 66, 505-529.

12. Mastronarde, D.N. (1987 a) Two classes of singlc-inpul X-cclls in cat lateral geniculate nucleus. I. Receptive field properties and classification of cells. J. Neurophysiol. 57, 357-380.

13. Mastronarde, D.N. (1987 b) Two classes of single-input X-cells in cat lateral geniculate nucleus. II. Retinal inputs and the generation of receptive field properties. J. Neurophysiol. 57, 381-413.

14. Saul, A.B, & Humphrey, A.L. (1990). Spatial and temporal response properties oflaggcd and non-lagged cells in cat lateral geniculate nucleus. J. Neurophysiol. 64, 206-224.

15. Wimbauer, S., Wcnisch O.G., Miller K.D., Hemmen, J,L, van (1997a) Development of spatiotcmporal receptive fields of simple-cells: I Model formulation. Biol. Cyber. 77,453-461.

16. Jagadccsh B., Wheat H.S. & Fcrstcr D. (1993) Linearity of summation of synaptic potentials underlying direction seiectivity in simple cells of cat visual cortex Science 262: 1901-1904.

17. DcAngelis, G.C., Ohzawa, I. & Freeman, R.D. (1993a) Spatiotemporal Organization of simple-cell receptive fields in the cat's striate cortex. I. General characteristics and postnatal development. J. Neurophysiol. 69, 1091-1117.

18. DcAngelis, G.C., Ohzawa, I. & Freeman, R.D. (1993b) Spatiotemporal Organization of simple-cell receptive fields in the cat's striate cortex. II. Linearity of temporal and spatial summation. J. Neurophysiol. 69,1118-1135.

19. Volgushev, M., Vidyasagar, T.R. & Pei, X. (1995) Dynamics of orientation tuning of post synaptic potential in cat visual cortex.

Visual Neurosci, 12,621-628.

20. Worgottcr, F. & Koch, C. (1991) A detailed model of (he primary visual pathway in the cat: Comparison of afferent excitatory and intracortical inhibitory connection schemes for orientation selectivity. J. Ncurosci. I I , 1959-1979.

21. Bhaumik B, and Mathur M, 2001 Proc. UCNN, Washington DC,

! 5-19 July, Vol 1,284-289.

22. Miller, K.D. (1994) A model for the development of simple ceil receptive fields and the ordered arrangement of orientation columns through activity dependent competition between ON and OFF center inputs. J. Neurosci. 14, 409-441.

23. Maex, R. & Orban, G.A. (1996) Model circuit of spiking neurons generating directional selectivity in simple cells. J. Neurophysiot.

75, 1515-1545.

24. Dong, D.W., Atick, J.J. (1995) Temporal dccorrclation: A theory of lagged and non-Sagged responses in the lateral gcniculatc nucleus. Networks, 159-178.

25. Gcrstncr, G. (1999) Spiking Neurons, in Mass W., & Bishop, C M . , eds., Pulsed Neural Networks, MIT Press, Cambridge.

26. Turrigiano G.G. & Nelson S.B. (1998) Thinking globally, acting locally: AMPA receptor turnover and synaptic strength.

Neuron 1998 Nov 21: 933-5