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Dependence of Response Properties on Sparse Connectivity in a Spiking Neuron Model of the Lateral Geniculate Nucleus

Laboratory for Intelligent Imaging and Neural Computing, Department of Biomedical Engineering, Columbia University, New York, New York

Submitted 14 June 2007; accepted in final form 15 September 2007

	ABSTRACT

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS REFERENCES

 
We present a large-scale anatomically constrained spiking neuron model of the lateral geniculate nucleus (LGN), which operates solely with retinal input, relay cells, and interneurons. We show that interneuron inhibition and sparse connectivity between LGN cells could be key factors for explaining a number of observed classical and extraclassical response properties in LGN of monkey and cat. Among them are 1) weak orientation tuning, 2) contrast invariance of spatial frequency tuning in the absence of cortical feedback, 3) extraclassical surround suppression, and 4) orientation tuning of extraclassical surround suppression. The model also makes two surprising predictions: 1) a possible pinwheel-like spatial organization of orientation preference in the parvo layers of monkey LGN, much like what is seen in V1, and 2) a stimulus-induced trend (bias) in the orientation and phase preference of surround suppression, originating from the stimulus discontinuity between center and surround gratings rather than from specific circuitry.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS REFERENCES

 
The lateral geniculate nucleus (LGN) constitutes an important processing stage in the early visual pathway and is the major source of afferent sensory input into the primary visual cortex (V1). Compared with V1, however, surprisingly little effort has gone into developing anatomically and physiologically constrained models of the LGN. One possible reason is that the LGN is perceived to be "less interesting" because its anatomy and its responses to visual stimulation are in many respects simpler than those of V1. However, the attraction for modeling studies is that several visual response properties are shared by LGN and V1, with some of them appearing in less pronounced form in LGN than in V1. Given the dazzling complexity of V1, the simpler anatomy of the LGN and its shared response properties with V1 make it an interesting object for modelers to explore in terms of identifying mechanisms underlying early vision.

Among the visual responses shared by LGN and V1 are classical orientation tuning, which is substantially weaker in LGN than in V1 (Shou and Leventhal 1989; Shou et al. 1986; Smith et al. 1990; Sun et al. 2004; Xu et al. 2002), and spatial frequency tuning, which is also somewhat weaker in LGN (mostly low-pass) than in V1 (mostly band-pass) (Hicks et al. 1983; Irvin et al. 1993; Kaplan and Shapley 1982).

LGN and V1 also share several extraclassical response properties. Among them are surround suppression (length tuning) and contrast dependence of the receptive field (RF) size (Anderson et al. 2001; Cavanaugh et al. 2002a; Dow et al. 1981; Felisberti and Derrington 1999, 2001; Jones et al. 2000; Kapadia et al. 1999; Kruger 1977; Levick et al. 1972; Ozeki et al. 2004; Sceniak et al. 1999, 2001, 2006; Schiller et al. 1976; Silito et al. 1995; Solomon et al. 2002). Notably, it has been found that extraclassical surround suppression in LGN is comparable in strength to what is observed in V1, whereas receptive field expansion for low contrast is somewhat less in LGN than in V1.

There is some experimental evidence for at least a partial transfer of extraclassical surround suppression from LGN to V1 (Ozeki et al. 2004; Webb et al. 2005). As of late, further experimental verification of this and to what extent this in fact takes place have been receiving more attention. In our previous work on extraclassical phenomena we also argued, based on a demonstration of feasibility by simulation, that some extraclassical responses in V1 are partially transferred from LGN (Wielaard and Sajda 2006b). Irrespective of whether transfer of extraclassical responses in fact occurs, understanding of the classical and extraclassical responses in LGN is necessary in its own right. In addition, it will help in understanding of the mechanisms governing these responses in V1.

In this study we present a large-scale spiking neuron model of the LGN. We explore what can be achieved in terms of response properties by modeling only the retinal input and neural connectivity between interneurons and relay cells within LGN, while ignoring other inputs such as those from cortex and brain stem. One might argue that this approximation is a rather drastic one. Particularly, in synaptic terms, cortical feedback is well known to be substantial (e.g., Sherman and Guillery 1996; van Horn et al. 2000). However, it is equally well known that retinal ganglion cells are dominant in driving responses in LGN (e.g., Reid and Shapley 1992; Usrey et al. 1999). A model like ours, neglecting feedback of any kind, is thus not an unreasonable approximation. These issues are addressed in further detail in the DISCUSSION section.

There are several motivations for studying an LGN model that considers only the feedforward pathway, rather than a model that attempts to address relative contributions from feedforward and feedback connections. One of course is clarity and transparency. Moreover, given our current knowledge of LGN and cortex, addressing these issues on a purely theoretical basis seems simply not yet feasible. Meaningful answers must necessarily be derived from the outcome of carefully designed experiments.

The parameters of our model are further anatomically and physiologically constrained by relevant data for the magno and parvo cellular layers of macaque and the X-cell network of layer A in cat. We demonstrate that in this way we are able to obtain a variety of classical as well as extraclassical responses, in good agreement with experimental data. Also, we are able to explain some characteristic differences observed in experimental data taken from monkey and cat. By analysis of the neural mechanisms underlying the response properties, we demonstrate that the sparseness of the connectivity, as determined by the length scales of intergeniculate connections, is a key parameter in setting the classical and extraclassical responses of our model LGN.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS REFERENCES

 
Model summary

We provide a brief description of the LGN model. The model is similar in spirit to our recently developed cortical model (see Wielaard and Sajda 2006a,b).

The model consists of a two-dimensional (2D) sheet of LGN cells and a 2D sheet of retinal ganglion cells. The retinal ganglion cells provide the sole input to our LGN cells; we ignore all other inputs (from cortex, brain stem, etc.). The model includes recurrent inhibitory connections among the LGN cells; it does not include recurrent connections among the retinal ganglion cells.

Following experimental data, the LGN cell population in the model is made up of 75% excitatory cells (relay cells) and 25% inhibitory cells (interneurons) (LeVay and Ferster 1979; Madarasz et al. 1986; Montero 1986; Montero and Zempel 1986). The cells are distributed randomly (and independently) on a square lattice. Experimental data show that, in both cat and monkey, the receptive field of an LGN cell closely resembles the receptive field of just one dominant retinal ganglion cell or that of several retinal ganglion cells with strongly overlapping receptive fields (Cleland and Lee 1985; Cleland et al. 1971; Lee et al. 1983; Reid and Shapley 1992; Usrey et al. 1999). Therefore—and also given the 2D nature of the model—the connectivity between retinal ganglion cells and LGN cells is taken to be one to one for simplicity, i.e., each LGN cell (including interneurons) receives input from its corresponding retinal ganglion cell.

We configured the model for the magno and parvo layers of macaque LGN, as well as for the X-cells in layer A of cat LGN, at parafoveal eccentricities (<5° for macaque, <10° for cat). For the magno macaque and X cat simulations, retinal ganglion cells are randomly taken to be either ON or OFF with an equal number of both cell types. For the parvo macaque simulations all retinal ganglion cells are taken to be of one type (ON or OFF), consistent with anatomical observations (Schiller and Malpeli 1978).

Retinal ganglion cell responses are modeled as rectified center–surround difference-of-Gaussian (DOG) spatiotemporal linear filters

(1)

Here [x]₊ = x if x 0 and [x]₊ = 0 if x 0, (r) and G_j^RGC() are the spatial and temporal retinal ganglion cell kernels, respectively; _j is the receptive field center of the jth retinal ganglion (LGN) cell; and I(, s) is the visual stimulus. The parameters g_j⁰ represent the maintained activity of retinal ganglion cells and the parameters _j^V measure their responsiveness to visual stimulation. The g_j⁰ values are drawn randomly and independently from a uniform distribution between 20 and 25 s^–1, and _j^V = 10 cd^–1. For the retinal ganglion cell kernels we used

(2)

and

(3)

where k is a normalization constant; _c and _s are the center and surround sizes, respectively; and K is the integrated surround–center sensitivity. Note that the spatial kernel is (for a given configuration) not cell specific; other than that, it is ON (+) or OFF (–).

Because we consider here only steady-state responses (drifting grating stimuli), we ignored the detailed differences between the temporal kernels (impulse response) for macaque mango and parvo and cat X-cells (Benardete and Kaplan 1999a,b; Reid and Shapley 2002; Usrey et al. 1999). For all configurations we used the time constants ₁ = 2.5 ms, ₂ = 7.5 ms, and c = (₁/₂)⁶, where the delay times _j⁰ are taken from a uniform distribution between 10 and 20 ms. The temporal kernels are normalized in Fourier space, _– |^RGC()|d = 1, _j^RGC() = (2)^–1 _–G_j^RGC(t)e^–it dt, and are fully transient, i.e., _j^RGC(0) = 0.

For center and surround sizes we used _c values of 0.1, 0.04, and 0.25° (centers) and _s values of 0.5, 0.32, and 1.25° (surrounds), for the macaque mango and parvo and cat X-cell configurations, respectively. The integrated surround–center sensitivity was set to K = 0.55 (Croner and Kaplan 1995; Derrington and Lennie 1984; Hicks et al. 1983; Linsenmeier et al. 1982; Shapley 1990; Spear et al. 1984).

As retinogeniculate mapping we use the identity mapping plus a small scatter

(4)

where x_j is the LGN cell's spatial location, µ is a geniculate magnification factor, and _j are random vectors, with components drawn randomly and independently from the uniform distribution on [–a, a], where a = 0.7_c. The total number of cells was the same for all model configurations and equal to N = 4,096. For magno and parvo configurations we used data on cell densities in the LGN layers and in visual space (Conolly and van Essen 1984; Malpeli et al. 1996) to compute the sizes (L x L) and µ values for the different LGN models, yielding L_M = 2.4 mm, µ_M = 0.75 mm/deg and L_P = 1.6 mm, µ_P = 1.23 mm/deg, respectively. For the cat X-cell configuration we used a magnification factor µ_X = 0.3 mm/deg (Sanderson 1971a,b) and an average grid spacing for retinal X-cells of L_X(µN)^–1 = 0.125° (Peichl and Wässle 1983; Wässle et al. 1981), yielding L_X = 2.4 mm.

Dynamic variables of a model LGN cell i are its membrane potential v_i(t) and its spike train _i(t) =_k (t – t_i,k), where t is time and t_i,k is its kth spike time. The membrane potential and spike train of each cell obey a set of N equations of the form

(5)

These equations are integrated numerically using a second-order Runge–Kutta method with 0.1-ms time step. Whenever the membrane potential reaches a fixed threshold level v_T it is reset to a fixed reset level v_R and a spike is registered. The equation can be rescaled so that v_i(t) is dimensionless and C_i = 1, v_L = 0, v_E = 14/3, v_I = –2/3, v_T = 1, v_R = 0, and conductances (and currents) have dimension of inverse time.

The quantities g_E,i(t, _E) and g_I,i(t, [], _I) are the excitatory and inhibitory conductances of cell i. The notation _E(I) stands for external noise, and []_I stands for the spike trains of all (inhibitory) interneurons connected to cell i. We assume noise, interactions with interneurons, and retinal ganglion cell input act additively in contributing to the total conductance of a cell

(6)

The terms _E,i(t) and _I,i(t) are external stochastic contributions and are subsequently given. The terms g_I,i^LGN(t, []_I) are the contributions from the (inhibitory) interneurons and include only isotropic connections

(7)

where (I) denotes the population of interneurons. The functions G_I,j() describe the synaptic dynamics of the interneuron synapses and the functions _I,i(r) describe the strength and spatial range of the interneuron interaction with cell i. We assume the availability of postsynaptic sites N_d on a cell (dendrites) to decay exponentially as a function of distance with length scale D, i.e., N_d exp[–(r/D)²], and make a similar assumption for the presynaptic sites N_a (axons of interneurons), N_a exp[–(r/A)²]. The spatial coupling strength (assuming individual synapses have equal strength) between two cells then decays exponentially with length scale _eff² = D² + A² and can be written as

(8)

where r_i,j = ||_i – _j||, and with the normalization constant

(9)

In this way, the parameters c_I,P are interaction strengths that define the density and length scale invariant contribution of the interneuron population (I) to the conductance of an interneuron itself (P = I) or a relay cell (P = E). Their numerical values are c_I,E = c_I,I = 2. The change in membrane potential of cell i (P) due to a single spike of interneuron j (I) is proportional to c_I,P(_eff)^–2(n_I)^–1 exp[–(r_i,j/_eff)²], where n_I is the cell density of the interneuron population.

The synaptic temporal kernels G_I,j() are normalized to unity, _– G_I,j()d = 1, and are of the form

(10)

The kernels G_I,i have a fast [-aminobutyric acid (GABA)] component set by a_i, chosen from a uniform distribution between 3 and 6 ms, and a slow component (Gibson et al. 1999) defined by b = 10 ms, whereas = 3/2. The constants k_i are normalization constants. These kernels imply a spike memory of the order of 50 ms for the interneuron inhibition.

The external stochastic terms _µ,i(t) in Eq. 6 are given by

(11)

where the kernels G_I,i^P have the same form as Eq. 10, the kernel G_E,i^P is as given in Wielaard and Sajda (2006b), and _µ,i^P are Poisson spike trains [mean firing rates 100 spikes/s (µ = E) and 125 spikes/s (µ = I)] belonging to neuron i (different ones for each cell). The noise strengths _E,i⁰ are drawn from a uniform distribution between 1 and 6, and _I,i⁰ are drawn from a uniform distribution between 0 and 10.

We obtained estimates of the effective interaction length scales _eff for the different configurations from available experimental data (Bickford et al. 1999; Michael 1988; Robson 1993; Sherman and Friedlander 1988; Wilson 1989). Simulations were performed for two different length scales (min–max estimates) for each configuration. The different models are referred to as M1, M2; P1, P2; and X1, X2 for magno, parvo, and cat configurations, respectively. The effective length scales used in the different models are: for the magno models _eff = 0.2 mm (M1) and 0.4 mm (M2); for the parvo models _eff = 0.075 mm (P1) and 0.15 mm (P2); and for the cat X-cell models _eff = 0.1 mm (X1) and 0.2 mm (X2).

We did not include triadic circuitry (see, e.g., Sherman and Guillery 1996) explicitly in the model, i.e., in a synaptic fashion. However, with the model's circuitry as set, triadic interactions occur entirely spontaneously and are numerous in the model. For about 40% of the relay neurons, the circuitry is such (by chance) that the receptive field (RF) of its retinal ganglion cell overlaps for >93% with the RF of at least one retinal ganglion cell (of the same sign, ON or OFF) belonging to a nearby (_eff) interneuron. Recall we have a one to one mapping between ganglion cells and LGN cells, including the interneurons. From a perspective of the visual input, such a relay cell will thus receive triadic interactions in the sense that it will be excited as well as inhibited by the same local visual stimulation. Another motivation for not including triads explicitly on the synaptic level (i.e., triadic synapses) is that our aim in this study is to address sparsity in the connectivity. To this end, it is desirable to keep the circuitry as isotropic as possible without contradicting anatomical data, to properly address the effects of sparsity rather than of specific circuitry.

Stimuli and data collection

All experiments were performed with drifting grating stimuli, with luminance given by I(, t) = I₀[1 + cos (t – ·)], and average luminance I₀, contrast , temporal frequency , and spatial wave vector . We used a temporal frequency of 8 Hz in all simulations, which is close to the averaged preferred temporal frequencies of the model configurations. Unless varied as part of the experiment, the spatial frequency of all gratings was kept fixed and equal to 2, 4, and 1 c/deg for the M, P, and X configurations, respectively. Each stimulus was presented for 3 s and preceded by a 1-s blank stimulus. The procedure was repeated five times with different initial conditions and noise realizations. SEs in cycle-trial–averaged responses and conductances are negligible. Experiments were performed at "high" contrast, = 1, and "low" contrast, = 0.3.

Classical orientation tuning curves were obtained using large size drifting gratings, seven- to tenfold the average receptive field size. Orientation and direction selectivity are characterized by respectively the orientation index (OI) and direction index (DI)

(12)

(13)

Here r() is the response and is the orientation. Smaller OI (DI) indicates a lesser orientation (direction) selectivity. Purely symmetric responses [i.e., r( + ) = r()] have DI = 0 but can have arbitrary OI values. Responses independent of orientation (no selectivity) have OI = DI = 0. If the response differs from zero for only one orientation (maximum selectivity) then OI = DI = 1. If the response differs from zero for only two orientations apart then OI = 1 and DI is arbitrary.

Spatially averaged responses are obtained by averaging responses of the cells in 0.06-mm patches. For such spatially averaged responses the preferred orientation _p and the orientation index OI were computed, yielding the orientation and orientation selectivity maps shown in Figs. 3 and 4. The gradient = (_x, _y) of the orientation map is defined as

(14)

with a similar expression for _yP(i, j).

View larger version (57K): [in this window] [in a new window]   FIG. 3. Coarse-grained spatial distribution of orientation preference in the model lateral geniculate nucleus (LGN) for the M1(X2), M2, P1, and P2 configurations. Color coded is the preferred angle (P) of the combined response of neighboring cells. Images show regions of steady change in P, singularities, fractures, and saddles. Regions of fractures and singularities are bounded by red contours. Singularities appear as small black patches. Less fractured singularities are indicated by + when clockwise and x when counterclockwise. Note the large number of singularities (pinwheels) for the P2 case. Both cases, magno (M1, M2) as well as parvo (P1, P2), show an increasing singularity density with decreasing sparsity.

View larger version (72K): [in this window] [in a new window]   FIG. 4. Spatial distribution of orientation selectivity in the model. OI of individual cells is color coded. Explanation of contours is as in Fig. 3. Note that regions of sharper tuning (blue) anticorrelate with regions of fractures (bounded by red contours).

(15)

where k_high is the spatial frequency greater than preferred, at half the maximum response, and k_low is the spatial frequency less than preferred, at half the maximum response. For low-pass cells, i.e., cells that do not have a k_low, we set k_low equal to the smallest spatial frequency used. In this way practically all cells with bandwidths >4 are low-pass cells.

For the analysis of surround suppression and contrast-dependent receptive field size the drifting grating was confined to a circular aperture of varying radius r_A. Other parameters of the grating were kept fixed. Simulations were performed for about 25–35 different aperture centers _eff apart and confined to the central region (larger than _eff removed from the boundary) of the models. Samples for analysis consist of cells with receptive field centers <_s/20 away from the aperture center. (We assumed an LGN cell's receptive field center to coincide with the corresponding retinal ganglion cell's receptive field center.) Selected cells have preferred spatial frequency less than the grating's spatial frequency, preferred temporal frequency within 2 Hz of the grating frequency (8 Hz), and a maximum response at low contrast that is >(f_b + 5), where f_b is the mean blank response in spikes/s. In this way we collect about 70–80 cells in a sample, with approximately uniformly distributed preferred angles.

Surround suppression is characterized by comparing the neuron's maximum firing rate to its steady firing rate for large apertures. We define the receptive field size r as the minimum aperture radius for which the response f(r_A) is >95% of its maximum. We define the surround size R as the minimum aperture radius >r for which the suppression f_s(r_A) = f_max – f(r_A) is >95% of its maximum. We define the asymptotic response f as the average response beyond R. We define the suppression index SI as the relative surround suppression

(16)

where f₀ is the response to a blank stimulus. The suppression index SI is similar to the one used in Solomon et al. (2002).

Neural mechanisms in the model are analyzed based on spike responses and conductances. To a good approximation (see Wielaard and Sajda 2006b) the relation between instantaneous firing rate (t) and the cycle-trial–averaged excitatory and inhibitory conductances is given by a rectified weighted difference

(17)

with cell-dependent, but stimulus- and time-independent, gain >0 and threshold .

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION GRANTS REFERENCES

 
We performed simulations for six different model configurations corresponding to the magno (M) and parvo (P) layers of macaque and X-cells in layer A of cat LGN, with two different characteristic length scales each. The different model configurations are referred to as M1, M2; P1, P2; and X1, X2, respectively (see METHODS). The different length scales _eff are, respectively, 0.2, 0.4; 0.075, 0.15; and 0.1, 0.2 mm, and were taken from experimental data and represent min–max estimates. The different model configurations differ in the sparseness of their connectivity. This sparsity can be expressed by the dimensionless parameter = 1/(_eff²), where is the cell density and with larger indicating sparser connectivity. For the M1, M2; P1, P2; and X1, X2 configurations the parameter has the value 1/28, 1/114; 1/9, 1/36; and 1/7, 1/28, respectively. The M2 case thus has the least sparse connectivity; the P1 and X1 cases have the most sparse connectivity; and the M1, P2, and X2 cases have intermediate, about equally sparse connectivity.

We note that the visual sparsity is about equal for M, X, and P configurations. Visual sparsity can be expressed by the dimensionless parameter _V = 1/(_Vc²), where _V is the retinal ganglion cell density and _c is the center size (see METHODS). For all cases the visual sparsity is _V 1/4. Further, the dimensionless receptive field scatter is identical for all configurations and equal to 70% of the center size (see METHODS).

Differences between the different cases other than sparseness of the connectivity have deliberately been kept minimal to enhance transparency in the interpretation of the results. In fact, other than sparseness of connectivity, the only relevant difference between M, X, and P configurations is the fact that the M, X cases contain a 1:1 mixture of ON and OFF cells, whereas the P cases contain only one type, either ON or OFF. The most notable compromise made in this respect is that the retinal ganglion cell temporal kernels (impulse response) are identical for all cases. We note that this is acceptable only because we limit ourselves in this study to stationary responses to drifting grating stimuli.

From this modeling perspective, we note also that the M1 and X2 cases are in fact identical up to a trivial scaling factor in the visual field, i.e., the visual length scale of X2 is simply a factor 2.5-fold the visual length scale of M1. Thus covered by essentially the same simulation, the M1 and X2 cases do, however, represent approximations of quite different realities. M1 represents a macaque LGN magno layer with estimated maximally sparse connectivity, whereas X2 represents a layer A of cat LGN with estimated minimally sparse X-cell connectivity.

In what follows we discuss several classical as well as extraclassical response properties observed in the LGN models and identify their neural mechanisms. The discussion of classical response properties is useful in its own right. It also serves to add context and meaning to the discussion of extraclassical response properties, as noted in Wielaard and Sajda (2006b). We address the difference in behavior between the M, X, and P models and the function of sparseness of connectivity as expressed by the parameter . We demonstrate that interneuron inhibition and sparseness of connectivity could be key ingredients in the explanation of classical and extraclassical response phenomena in monkey and cat LGN and why the phenomena quantitatively differ in these animals.

Classical responses

ORIENTATION TUNING. Cells in the model LGN show weak orientation and direction selectivity for large drifting grating stimuli, in agreement with experimental data (Shou and Leventhal 1989; Shou et al. 1986; Smith et al. 1990; Sun et al. 2004; Xu et al. 2002). Tuning curves for several model cells and the distributions of orientation and direction index over all model cells are shown in Fig. 1. The spatial frequency of the drifting grating was 2 c/deg and temporal frequency was 8 Hz. These results are for the M1(X2) and M2 configurations of the model (see METHODS). Results for the P1, P2, and X1 configurations are qualitatively similar in that we observe about equal orientation selectivity and direction selectivity for P2 as for M1(X2) and, on average, roughly a doubling of these properties for P1 and X1. We thus observe a consistent increase in orientation and direction selectivity for increasing sparsity in the six model configurations.

View larger version (24K): [in this window] [in a new window]   FIG. 1. Orientation tuning in the M1(X2) and M2 models. Top: examples of orientation tuning curves for three model cells (I, II, III), with different orientation (OI) and direction (DI) indices. Plotted are average firing rates (F1). Horizontal arrows indicate background firing rate. Bottom: distributions of OI and DI. Vertical arrows indicate means, dashed (solid) arrow for unfilled (filled) histograms. M1(X2) and M2 models have different sparseness of connectivity. Tuning is caused by sparseness of connectivity and is seen to decrease with decreasing sparseness, i.e., for eff = 0.4 mm. Decrease is significant (Wilcoxon, P < 0.001 for OI and P < 0.000001 for DI).

That the orientation and direction selectivity observed indeed originates in the sparseness of the connectivity is illustrated in Fig. 2, in which plots (top) are shown for the tuning curves of cell I for the two levels of connectivity sparsity set by the inhibitory length scales _eff values of 0.2 and 0.4 mm. The responses are plotted for 16 grating orientations from – to 7/8. In the lower section the cell's excitatory and inhibitory conductances are plotted for these 16 orientations. Apart from a trivial phase factor, the excitatory conductance g_E(t) of the cell is independent of the grating orientation. This is true for all cells in the model because there is no recurrent excitation. Excitation arises solely from the retinal ganglion cell inputs, which are given by a center–surround convolution with the stimulus and thus rotationally symmetric, apart from a phase factor. Another observation from Fig. 2, which we find to hold in general, is that the mean of the inhibitory conductance g_I(t) is nearly insensitive to the grating orientation (<5% change).

View larger version (32K): [in this window] [in a new window]   FIG. 2. Illustration of the mechanism of orientation (direction) tuning for cell I in Fig. 1 for the 2 different degrees of sparseness: eff = 0.2 mm (M1, X2) and eff = 0.4 mm (M2). Spike responses (F0) are plotted in the top. Corresponding cycle-trial–averaged excitatory and inhibitory conductances for each grating orientation are plotted in the middle. Orientation (direction) tuning is seen to arise from phase differences in the modulations of the conductances and not from changes in their means. This is illustrated in more detail in the bottom.

Figure 2 is just one example of a directionally selective cell in the model. As shown in Fig. 1, we also observe a considerable diversity in orientation and direction selectivity in the model. This diversity can be intuitively understood by using a simple linear approximation to the full nonlinear model. Such an approximation is obtained when we assume that the inhibitory conductance of a particular cell n is simply proportional to the sum of the ganglion cell inputs into all interneurons within a distance _eff. That is, if _n indicates this set of interneurons, we assume that

(18)

where A_m, B_m > 0. The phase factor _m() depends on the temporal delay of the retinal ganglion cell as well as the spatial position of its receptive field, and it has a more or less random behavior for a population of cells. Recall (see METHODS) that, neglecting noise, we also have

(19)

Equations 18 and 19 illustrate several points. First, the amplitude of the waveform resulting from the summation in Eq. 18 will in general be larger when the sum contains fewer terms, i.e., for larger sparsity. Second, whatever waveform results from the summation in Eq. 18, it will in general have a different dependence on the grating orientation than that in Eq. 19. Thus we may in general expect to find some orientation/direction selectivity in the model. Finally, the resulting waveform g_I,n(t) depends on the set of neighboring interneurons _n, which is a different set for each cell n, so we may also expect diversity in the orientation/direction selectivity.

Next we turn to the spatial organization of orientation tuning in the model. Figure 3 shows the coarse-grained spatial organization of the preferred orientation of the model configurations M1(X2), M2, P1, and P2. The preferred angles _P of the combined response of nearest-neighbor cells within a 30-µm radius are color coded. Interestingly, the images show a behavior similar to that observed in V1 (Blasdel 1992a,b; Obermayer and Blasdel 1993), that is, regions of steady change in _P, singularities (pinwheel centers), fractures, and saddles. Red contours indicate boundaries of regions where ||_P|| >0.45 (fractures; see METHODS). Singularities appear as small black patches, which are regions where ||_P|| >0.75. Unlike in our V1 model (Wielaard and Sajda 2006b), the orientation maps in Fig. 3, however, appear entirely spontaneous—no particular spatial structure is present in the input. The orientation maps in the LGN model arise from spontaneous dynamic organization (self-organization) of the interneuron inhibition. This is also apparent from the observed trends in the orientation maps, with the maps becoming more organized (more pinwheels) for longer-range inhibition. As can be seen in Fig. 3, we observe approximately a doubling of the number of pinwheels when comparing M2 versus M1(X2) and P2 versus P1. In contrast with what we found for orientation/direction selectivity, it is not primarily sparsity that controls the spatial structure of the orientation map. Randomness in the input starkly hinders the self-organization. Orientation maps are better organized when less randomness is present in the input. This is evident from Fig. 3: the configurations M1(X2) and P2, which have about equal sparsity, differ greatly in their orientation map, with P2 showing a much better organization. Recall that the primary difference between the two cases is that M1 contains a balanced mixture of ON and OFF cells, whereas P2 contains cells of only one type, which means that the M1(X2) case has a more random input than P2, and this can be seen with the help of Eq. 19. Changing cell n from an ON cell to an OFF cell (or vice versa) simply implies adding an extra phase factor to it _n(). Thus doing this for half of the cells will broaden the distribution of _n() over the cell population.

Finally, we discuss the spatial distribution of orientation selectivity in the model. Plots of the spatial organization of the orientation index (OI) for the M1(X2) and P2 cases are shown in Fig. 4. In sharp contrast to the orientation map, the spatial distribution of orientation selectivity looks very similar for the M1(X2) and P2 cases. Because M1(X2) and P2 have about equal sparsity, this is in line with our earlier observation that orientation selectivity in the model depends primarily on the sparseness of the connectivity. Unlike what is observed in V1 (Blasdel 1992b), we do not observe a positive correlation between regions of sharper tuning (blue) and regions of fractures (enclosed by red contours). Rather, on the contrary, as can be seen in Fig. 4, we observe a negative correlation between the two. The explanation (not shown) is that fractures in the orientation map occur where inhibition is relatively less organized, i.e., relatively weak. Regions of smoothly varying orientation preference occur where inhibition is relatively well organized, i.e., relatively strong. Because orientation selectivity is entirely generated by the interneuron inhibition, and stronger inhibition generates higher selective cells, regions of better tuned cells will be located in regions of smoothly varying orientation preference, i.e., will anticorrelate with regions of fractures.

SPATIAL FREQUENCY TUNING. In this section we briefly discuss the model's behavior as a function of spatial frequency. Of interest by itself, it is also of relevance with regard to the extraclassical phenomena of surround suppression and receptive field expansion, discussed in the following sections.

As described in METHODS, we used a center–surround DOG model for the ganglion cell receptive fields. In the model, all ganglion cell receptive fields are identical in their spatial structure, up to a translation. Thus all ganglion cells have identically shaped spatial frequency tuning curves, differing only by a normalization factor (Eq. 1), and the same is true for the feedforward excitatory inputs in our LGN cells. The shape of this tuning curve is illustrated symbolically by the thick dotted curve in Fig. 5A.

View larger version (27K): [in this window] [in a new window]   FIG. 5. Summary of spatial frequency tuning properties in the model for the 2 magno configurations, M1 (eff = 0.2 mm) and M2 (eff = 0.4 mm). A and B: spatial frequency tuning curves (firing rate, first harmonic F1) obtained with a large, high-contrast 8-Hz drifting grating of varying spatial frequency for 4 model cells (same cells in both models), and for a retinal ganglion cell (thick dotted curve, drawn on arbitrary scale for clarity). C: distribution of the preferred spatial frequency kP. D: scatterplot of the spatial frequency bandwidth (β) at high and low contrast.

In contrast with orientation tuning, however, where we found a strong dependence of selectivity on sparseness, the spatial frequency selectivity is only weakly dependent on sparseness, and decreases only slightly for M2 (_eff = 0.4 mm) with respect to M1 (_eff = 0.2 mm). Note that using the results of Fig. 5, A and B, and of the previous section, we may conclude that, in agreement with experimental observation (Usrey et al. 1999), the center–surround DOG model will be a reasonable approximation for an LGN cell's receptive field, but with parameters slightly different from those for the corresponding ganglion cell.

The model's distribution of preferred spatial frequencies, shown in Fig. 5C, agrees reasonably well with experimental data (Hicks et al. 1983; Irvin et al. 1993; Kaplan and Shapley 1982). Because all excitatory inputs have identically shaped spatial frequency tuning curves (thick dotted curve in Fig. 5A), with preferred frequency of 1.05 c/deg, the diversity observed in the spatial frequency tuning is entirely due to the inhibitory interaction resulting from the interneurons. The distribution is seen to be rather insensitive to the sparsity level.

Figure 5D shows a scatterplot of the spatial frequency bandwidth β (see METHODS) for high- and low-contrast stimuli. The diversity in bandwidths in the model also shows good agreement with experimental data (Hicks et al. 1983; Irvin et al. 1993; Kaplan and Shapley 1982). Note that the small decrease in sharpening for decreasing sparsity is also apparent in this plot because the open circles (M2, _eff = 0.4 mm) are shifted slightly away from the origin with respect to the filled circles (M1, _eff = 0.2 mm). Also note the predominantly low-pass nature of the cells' spatial frequency tuning, in agreement with experimental observation.

Importantly, however, as illustrated in Fig. 5D, for both levels of sparsity we do not observe a statistically significant change in bandwidth as a function of contrast. A narrowing in bandwidth for lower contrast can be interpreted as the spatial frequency domain equivalent of contrast-dependent receptive field expansion. In both our V1 model (Wielaard and Sajda 2004, 2006b) and in experimental data (Nolt et al. 2004; Sceniak et al. 2002) a decrease in spatial frequency tuning bandwidth is observed to accompany low-contrast receptive field expansion. Contrast-dependent receptive field expansion has also been observed in LGN (Solomon et al. 2002); however, recent experimental observations (Sceniak et al. 2006) show that in the absence of feedback from V1, LGN cells show little or no contrast-dependent receptive field expansion. The results shown in Fig. 5D are thus consistent with these experimental data.

All results discussed in this section are for the M1 and M2 configuration. Spatial frequency tuning curves for the X2 case are of course identical to those for the M1 but shifted 1/2.5 c/deg to the left (on a log scale) and are in good agreement with experimental data from cat (Derrington and Fuchs 1979; Rodieck and Stone 1965; So and Shapley 1979). Qualitatively the comparison between X1 and X2 is similar to what we discussed here for M1 and M2. Cells in the P1, P2 configurations have higher preferred spatial frequencies (about twofold) than those in the M1, M2 cases, but bandwidths similar to those in the M1, M2 cases. Again, qualitatively, the relative comparison between P1 and P2 is similar to what is discussed for the M1 and M2 cases.

Extraclassical responses

SURROUND SUPPRESSION AND RECEPTIVE FIELD EXPANSION. As pointed out in Wielaard and Sajda (2006b), a center–surround DOG model, such as given by Eqs. 1–3 for the ganglion cell receptive fields, does not show surround suppression resulting from the classical surround, for drifting gratings with spatial frequencies equal to or greater than the preferred spatial frequency. At lower spatial frequencies such a model does show surround suppression caused by the classical surround, and we referred to this as classical surround suppression (see Wielaard and Sajda 2006b). At significantly higher spatial frequencies than preferred (roughly a factor 5) such a model shows surround suppression that is unrelated to the classical surround, but caused by resonance between the spatial frequency and the inverse of the center size (see Wielaard and Sajda 2006b).

Here we seek to address truly extraclassical surround suppression. This is achieved by using drifting gratings with spatial frequency about twofold (rounded to whole numbers) larger than the preferred spatial frequency of the DOG retinal ganglion cell model used. For example, the preferred spatial frequency for the retinal ganglion cells follows from a simple formula that for the M configurations yields 1.05 c/deg, whereas the grating frequency used for the M simulations is 2 c/deg. For the P and X simulations the grating spatial frequencies are 4 and 1 c/deg, respectively (preferred 1.89 and 0.42 c/deg). At these spatial frequencies the model's retinal ganglion cells do not show surround suppression. Thus for the stimuli used in our simulations, the surround suppression in the model is entirely generated by interneuron inhibition because there is no other source available by which it could occur. Note that the preferred spatial frequency of LGN cells differs in general from that of the ganglion cells and this difference is a result of the interaction with interneurons. All LGN cells selected for study had a preferred spatial frequency less than the grating frequency.

The experimental definition of extraclassical surround suppression requires that it is observable only when stimulation occurs simultaneously in the central part of the receptive field, the so-called classical receptive field. Stimulation of the extraclassical surround alone without stimulation of the classical receptive field yields no response. We note that the model's surround suppression is consistent with this definition (not shown, but see Wielaard and Sajda 2006b).

A demonstration that the surround suppression in the model indeed occurs solely by means of the interneuron inhibition is given in Fig. 6, for a representative M1 model cell. Plotted are the cell's response as a function of aperture size in the top panel and the corresponding conductances in the bottom panel. The drifting grating used had a spatial frequency of 2 c/deg and an 8-Hz temporal frequency. The cell's preferred spatial frequency is 1.1 c/deg. We see that the excitatory conductance g_E saturates one aperture after the aperture of maximum response (receptive field size). The excitatory conductance arises solely from the ganglion cell input (and noise; see METHODS) and thus shows no surround suppression after saturation. The inhibitory conductance g_I is seen to continue its increase well after the aperture of maximum response, and this is causing the surround suppression in the cell's response. That the inhibitory conductance saturates more gradually (as a function of aperture size) than the excitatory conductance originates from the fact that it is generated by neighboring (_eff) interneurons that have offset receptive fields with respect to the receptive field of the cell studied. Thus it takes a larger aperture to saturate the excitatory drive of the relevant interneurons than to saturate the excitatory drive of the cell studied.

View larger version (21K): [in this window] [in a new window]   FIG. 6. Surround suppression by interneuron inhibition. Top: spike responses (F0) are plotted for high and low contrast. Bottom: for high contrast, corresponding cycle-trial–averaged excitatory and inhibitory conductances for each aperture radius are plotted. Surround suppression is seen to arise from the increase in inhibition with increasing aperture radius. As explained in the text, the excitatory conductance gE(t) (i.e., retinal ganglion cell input) is seen to show no surround suppression (bottom). This is also illustrated by the gray dashed curve in the top [maximum of gE(t) as a function of aperture size]. Same holds for low contrast (not shown). Surround suppression decreases for low contrast with no notable change in the receptive field size.

A summary of our results for surround suppression and contrast-dependent receptive field expansion for the M1 case is provided in Fig. 7. This sample consisted of 78 cells. The drifting grating used had spatial and temporal frequencies of 2 c/deg and 8 Hz. Information on how we selected the cells in the sample, contrast levels, definitions of receptive field size, surround size, and suppression index SI are given in METHODS. Briefly, the receptive field size is the smallest aperture of maximum response, the surround size is the smallest aperture of maximum suppression, and the suppression index is the relative suppression with respect to the maximum response.

View larger version (20K): [in this window] [in a new window]   FIG. 7. Summary of surround suppression and receptive field expansion in the M1 model. Sample consisted of 78 cells. Arrows indicate means, dashed (solid) arrow for unfilled (filled) histograms. A: distribution of receptive field size at high and low contrast. B: distribution of surround size at high and low contrast. C: distribution of low/high contrast ratio for receptive field and surround sizes. Growth in both receptive field size and surround size is statistically significant (Wilcoxon, P < 0.0001). D: distribution of surround suppression at high and low contrast. Decrease in surround suppression is significant (Wilcoxon, P < 0.000001). For the X2 configuration, results for growth ratios and suppression index are identical; distributions of receptive field and surround sizes shift by a factor of 2.5 to the right.

In agreement with experimental data (Sceniak et al. 2006; Solomon et al. 2002) we also observe a decrease of the average surround suppression for lower contrast, as shown in Fig. 7D. Note also that the shape of the SI distribution is centered around its mean, which agrees with the experimental observations, and differs from the shape of the suppression index distribution in V1, which is skewed toward SI = 0.

Results discussed are for the M1 configuration. The other configurations yield qualitatively similar results.

Orientation tuning of suppression. We studied the orientation and phase selectivity of surround suppression in the model using an aperture–annulus configuration of two drifting gratings, each with identical spatial and temporal (8 Hz) frequency and contrast. The spatial frequencies of the gratings are again set to about twice the preferred frequency of the retinal ganglion cells for each configuration, as explained earlier. The cell sample used for the analysis in this section is identical to that used to analyze surround suppression and receptive field expansion.

In the simulations, one of the gratings (orientation _C, phase _C) was confined to a centered aperture with radius 1.1-fold the sample averaged classical receptive field size. The second grating (orientation _S, phase _S) was confined to a concentric annulus with inner radius 1.5-fold the sample averaged classical receptive field size and outer radius of 3° for M and P simulations and 7.5° for X simulations. Parameters of the central grating were kept fixed; orientation and phase of the annulus grating were varied. We defined the surround suppression f_S as the difference between the response (mean firing rate, F0) when the central grating was presented alone and the response when it was simultaneously presented with the annulus grating. In general the surround suppression depends on the orientation and the phase difference between the center and surround gratings and on _C alone and not on _C alone, i.e., f_S = f_S(_C, _C – _S, _C – _S) = f_S(_C, , ). We find that for the model the dependence is predominantly on and and only weakly on _C alone. Measures of orientation, direction, and phase selectivity when referring to surround suppression are based on f_S (see METHODS).

We observe a rich diversity in surround orientation tuning for the M1(X2) model. Surround tuning curves (f_S as a function of ) of three model cells from this configuration are shown in Fig. 8. The responses are plotted for fixed _C and 16 surround grating orientations with ranging from 0 to 15/8. Cell A is characterized by a directionally selective surround suppression; maximum suppression occurs when surround and center grating have equal drift direction and minimal suppression occurs when they have approximately opposite drift direction. Cell B is characterized by orientation but not direction selective surround suppression; maximum suppression occurs when center and surround grating have approximately the same orientation and minimal suppression occurs when they have approximately orthogonal orientation. The surround suppression of cell C is nonselective for orientation. In the bottom sections of Fig. 8 the excitatory and inhibitory conductances of cells A and B are plotted for the 16 orientations in the top section.

View larger version (29K): [in this window] [in a new window]   FIG. 8. Orientation tuning of surround suppression in the M1(X2) model. Top: plots of surround tuning curves (F0) for 3 model cells. Bottom: plots of cycle-trial–averaged conductances for cells A and B at corresponding surround orientations. Notice the 3 properties of the inhibitory conductance that depend on the surround orientation: (i) the phase of the dominant modulation, (ii) the amplitude of the modulations, and (iii) the mean (F0) conductance. Notice that the excitatory conductance does not noticeably depend on the surround orientation.

Sparseness of the connectivity primarily causes temporal modulations in the cycle-trial–averaged inhibitory conductance g_I(t) (Fig. 8, bottom). These modulations depend, both in amplitude and phase, on the grating orientation. They are cell specific, as explained earlier, and are the major cause of the diversity in the orientation tuning of surround suppression in the model. Note in this respect that the excitatory conductance g_E(t) of the cell does not noticeably depend of the surround grating orientation; again, this is true for all cells in the model because there is no recurrent excitation and excitation arises solely from the retinal ganglion cell's center–surround inputs, in effect set by the central grating for the cell studied (and not by the surround grating).

The effect of stimulus discontinuity is subsequently addressed in detail. Combined with sparsity in visual space and in connectivity, it primarily creates a trend (noncell specific) in the dependence of the mean (F0) inhibitory conductances on the relative orientation and relative phase of center and surround grating.

Returning to Fig. 8 and comparing the responses of cell A at grating orientations = 0 and = 6/8, we see that around the maximum surround suppression ( = 0) g_I(t) has a dominant in-phase modulation with respect to g_E(t), whereas at the minimum suppression ( = 6/8) g_I(t) has a dominant antiphase modulation. We also see that the amplitude of the modulations plays a role in determining the suppression. Both amp