Abstract

To date, Hebbian learning combined with some form of constraint on synaptic inputs has been demonstrated to describe well the development of neural networks. The previous models revealed mathematically the importance of synaptic constraints to reproduce orientation selectivity in the visual cortical neurons, but biological mechanisms underlying such constraints remain unclear. In this study, we addressed this issue by formulating a synaptic constraint based on activity-dependent mechanisms of synaptic changes. Particularly, considering metabotropic glutamate receptor-mediated long-term depression, we derived synaptic constraint that suppresses the number of inputs from individual presynaptic neurons. We performed computer simulations of the activity-dependent self-organization of geniculocortical inputs with the synaptic constraint and examined the formation of receptive fields (RFs) of model visual cortical neurons. When we changed the magnitude of the synaptic constraint, we found the emergence of distinct RF structures such as concentric RFs, simple-cell-like RFs, and double-oriented RFs and also a gradual transition between spatiotemporal separable and inseparable RFs. Thus, the model based on the synaptic constraint derived from biological consideration can account systematically for the repertoire of RF structures observed in the primary visual cortices of different species for the first time.

1.  Introduction

Many lines of evidence that support the notion that N-methyl-D-aspartate (NMDA) receptor/channels work as a biological counterpart to the Hebbian learning rule have been accumulated (Kleinschmidt, Bear, & Singer, 1987; Gu, Bear, & Singer, 1989; Bear, Kleinschmidt, Gu, & Singer, 1990; Ramoa, Paradiso, & Freeman, 1988). That is, when postsynaptic potential is depolarized to alleviate Mg2+ from NMDA channels and glutamate released from a presynaptic terminal binds postsynaptic NMDA receptors at the same time, Ca2+ ions flow into the postsynaptic neuron. This Ca2+ influx induces a transient increase in intracellular [Ca2+] at the spine, which leads to the generation of long-term potentiation (LTP). When glutamate binds to NMDA receptors while the postsynaptic potential is at rest or hyperpolarized, no LTP is induced. Therefore, the coincidence of postsynaptic depolarization and the arrival of spike activity at the presynaptic terminal is the key step in LTP induction and spine outgrowth. However, this form of LTP mechanism alone cannot explain the elaboration of synaptic connections. In most mathematical models of the self-organization of synaptic connections based on the Hebbian learning rule, some constraints on synaptic connections are assumed to prevent the divergence of synaptic strengths or unrealistic synaptic wiring patterns (von der Malsburg, 1973; Linsker, 1986; Miller, Keller, & Stryker, 1989; Tanaka, 1990; Miller, 1994; Thomas & Cowan, 2006). In this sense, a synaptic constraint is as important as the Hebbian learning rule when we attempt to reproduce a physiologically plausible neural structure using a self-organization model. In this study, we reconsidered the possible mechanism of activity-dependent long-term depression (LTD) based on experimental findings on the roles of metabotropic glutamate receptors (mGluRs) in LTD induction and the effect of postsynaptic hyperpolarization. We obtained a mathematical form of synaptic constraint from such considerations, which limits the number of afferent inputs from active presynaptic neurons to a single postsynaptic neuron when presynaptic activation simultaneously occurs with postsynaptic inhibition. We examined possible roles of the constraint in receptive field (RF) formation in a simple model of many-to-one afferent inputs, in which local inhibitory circuits were implicitly considered to generate the synaptic constraint. Taking into account ON- and OFF-center cells in the lateral geniculate nucleus (LGN) with a short or long latency, we performed computer simulation of the model to examine the emergence of spatiotemporal RFs in the model cortical neuron under the presentation of drifting sinusoidal grating stimuli. Fourier analyses of self-organizing RFs demonstrated mainly six phases in RF structures for different magnitudes of constraint. Center-surround antagonistic, concentric RFs were formed for a very low magnitude of constraint. For a slightly increased magnitude of constraint, subfield-scattered RFs and spatiotemporal separable and inseparable single-oriented RFs emerged in intermediate magnitudes of synaptic constraint. Double-oriented RFs appeared for a high magnitude of constraint. An extremely strong constraint disrupted orientation selectivity, forming unoriented RFs. Neurons with concentric RFs appear in the thalamocortical recipient layer of the primary visual cortex of macaques (Blasdel & Fitzpatrick, 1984); however, most area 17 neurons in the corresponding layer of cats, minks, and tree shrews show oriented RFs (Hirsch, Alonso, Reid, & Martinez, 1998; McConnel & LeVay, 1986; Humphrey & Norton, 1980), while a small portion show double-oriented RFs (Worgotter & Eysel, 1991; Shevelev, Lazareva, Novikova, Tikhomirov, & Sharaev, 1994). Our simulation results for the first time reproduce the repertoire of RF structures of neurons in the recipient layer of the primary visual cortices of different species.

2.  Model Description

Our basic dynamical equation of afferent inputs from a presynaptic cell to a postsynaptic spine was given by the nonlinear Lotka-Volterra equation (Tanaka, 1990). Owing to strong competition among afferent inputs, its steady-state solutions describe a winner-take-all property indicating that only an afferent input is connected when its local energy is minimum, and the others are disconnected, where the local energy is given by the negative sum of the driving forces toward synaptic potentiation. This property enables mapping the connectivity of an afferent input to a binary variable, which takes 1 or 0. The nonlinearity of the system has been implicitly included by the binary variable derived from the winner-take-all property. Also, the noise effect leads to the thermodynamics of the system with the fictitious temperature β−1, which corresponds to the degree of synaptic plasticity. Thereby, in our framework, we have only to update synaptic connections using the Metropolis algorithm (Metropolis, Rosenbluth, Rosenbluth, Teller, & Teller, 1953), needless to solve the dynamic equation explicitly (Tanaka, 1990). Now we need to determine a specific configuration of pre- and postsynaptic neuronal layers and the driving forces for synaptic rewiring.

2.1.  Model LGN and Visual Cortex.

Here, we examined the formation of a spatiotemporal RF in a visual cortical neuron using a mathematical model that describes the activity-dependent rewiring of afferent inputs from the model LGN to a single visual cortical neuron. For simplicity, we omit intracortical recurrent connections to focus on the possible effects of synaptic constraint on the formation of geniculocortical inputs. According to anatomical studies of visual cortical neurons in the cat visual cortex (Wilson & Sherman, 1976; Gilbert & Wiesel, 1979; Martin & Whitteridge, 1984), the extent of dendritic arbors is distributed within the range of 200 to 400 μm. For computer simulation, we assume that the dendritic arbor of a model neuron is a disklike field whose diameter is discretized to 19 pixels, and hence the total number of dendritic lattice points is 261 pixels. When the lattice constant is set at 15 μm, the diameter of the dendritic field is given as 285 μm within the above-mentioned range of experimental observations. We assume that the LGN is composed of a retinotopically arranged 24 × 24 square lattice, and each of the lattice points contains four types of cells: nonlagged ON-center, nonlagged OFF-center, lagged ON-center and lagged OFF-center types (see Figure 1).

Figure 1:

Schematic diagram of the model. Two types of response latency, nonlagged and lagged, are assumed for each ON- or OFF-center cell in the model LGN. Each of the four types of model LGN cell is arranged retinotopically in a 24 × 24 square lattice, and one lattice point corresponds to one LGN cell. The dendritic field of a cortical neuron is represented by a circular disk whose diameter is 19 pixels. Each pixel receives a single afferent input from the model LGN. The distance between nearest-neighbor pixels in the dendritic field is set at 15 μm.

Figure 1:

Schematic diagram of the model. Two types of response latency, nonlagged and lagged, are assumed for each ON- or OFF-center cell in the model LGN. Each of the four types of model LGN cell is arranged retinotopically in a 24 × 24 square lattice, and one lattice point corresponds to one LGN cell. The dendritic field of a cortical neuron is represented by a circular disk whose diameter is 19 pixels. Each pixel receives a single afferent input from the model LGN. The distance between nearest-neighbor pixels in the dendritic field is set at 15 μm.

Next, we determine the driving forces for synaptic rewiring in the following subsections.

2.2.  NMDAR-Mediated LTP.

The transient influx of extracellular Ca2+ ions through NMDA channels is caused by postsynaptic local depolarization and the arrival of presynaptic spike activity at a spine of a postsynaptic neuron's dendrite. The intracellular [Ca2+], at time t at spine j that receives an afferent input from an LGN cell whose position k with cell type (μ1, μ2) is given by
formula
2.1
where ςDPj(t) and represent the local depolarization of a visual cortical neuron and the firing rate of the LGN cell at t, respectively. The subscripts μ1 and μ2 in cell type (μ1, μ2) represent ON-center inputs (μ1 = +1) or OFF-center inputs (μ1 = −1) and nonlagged inputs (μ2 = +1) or lagged inputs (μ2 = −1), respectively. The voltage-dependent alleviation of Mg2+ ions from NMDA channels requires that the membrane depolarization occurs immediately before the transmitter release at the LGN input. The kernel KNMDA(tt′) describes the temporal response of intracellular [Ca2+] at t to Ca2+ ion influx through NMDA channels.
When is used, the membrane depolarization ςDPj(t) is expressed as
formula
2.2
This [Ca2+] is enhanced by backpropagating action potential (bAP) elicited by a cortical neuron immediately before Ca2+ influx through the NMDA channels (Nevian & Sakmann, 2004). Consequently, the driving force for NMDAR-mediated LTP is given by
formula
2.3
where the brackets indicate the averaging procedure over time and stimulus patterns presented, and τ represents the delay of action potential propagation. The enhancement factor gCX(t)) is assumed to be given by gCX(t)) = 1 + q · ηCX(t), where q denotes a parameter determining the contribution of bAP to the increase in [Ca2+]. In the simulation, q is set to be the inverse of the time-averaged ηCX(t).
describes the dendritic propagation of activities, which is given by
formula
2.4
where denotes the distance between the dendritic sites j and j′, and the parameter λETL represents the electrotonic length of a dendrite. We set when the LGN cell (k, μ1, μ2) connects to the dendritic position j; otherwise we set . Using the function [x]+ = x for x ⩾ 0, 0 for x < 0, the firing rate ηCX(t) is given by the half-rectification of the membrane depolarization ζDP(t) at the soma with the threshold ϑth as
formula
2.5
where ζDP(t) is given as
formula
2.6
The firing threshold of a cortical neuron is given by ϑth = 〈ζCX(t)〉, which indicates a homeostatic regulation of firing rate depending on average membrane depolarization:
formula
2.7
The autocorrelation of spike activity should be zero in the case where the time shift is sufficiently smaller than the refractory period of an LGN neuron's activity. Owing to the small time shift ε, we can exclude (k, μ1, μ2) in the sum over (k′, μ′1, μ′2). Because the small time shift is absorbed by the jittering effect of spike generation at LGN neurons, we can exclude it. Also, if an LGN neuron's firing rate is not high enough, then only the membrane depolarization induced at other synapses can coincide with . Therefore, we obtain
formula
2.8
Assuming that the peak of the kernel is almost the same as the delay of somatic action potential propagating to the synaptic position, is approximately given by
formula
2.9
This term corresponds to the NMDAR-mediated Hebbian coincidence, which is enhanced by the backpropagation of action potential.

2.3.  Synaptic Constraint.

In young rats, a weak excitation of hippocampal CA1 neurons via Schaffer collaterals at a low frequency accompanied by repeated brief exposures to the inhibitory transmitter γ-aminobutyric acid (GABA) causes LTD of the stimulated pathway (Yang, Connor, & Faber, 1994). The induction of this form of LTD also requires an intracellular [Ca2+] increase. This conjunctive LTD is not blocked by the N-methyl-D-aspartate receptor (NMDA) antagonist but by an mGluR antagonist. Also, in the hippocampus of anesthetized adult rats, an activity-dependent LTD of glutamatergic transmission has been observed (Thiels, Barrionuevo, & Berger, 1994). When paired-pulse stimulation at short interstimulus intervals is associated with the inhibition of pyramidal cell firing on the second pulse of a pair, LTD is induced at stimulated inputs. This suggests that the inhibition of pyramidal cell activity is mediated by input to pyramidal cells from local GABAergic interneurons activated by commissural fibers or CA1 recurrent collaterals. Actually, such inhibition is eliminated by the local administration of the GABA receptor antagonist bicuculline near the recording site. Postsynaptic input from GABAergic interneurons is necessary for LTD induction, because short-ISI paired-pulse stimulation fails to induce LTD in the presence of bicuculline. However, NMDAR-mediated excitation is not absolutely needed, because APV does not abolish paired-pulse inhibition. Steel and Mauk (1999) have also demonstrated that the bath application of the GABAA agonist muscimol to hippocampal CA1 slices increases the range of frequencies that induce LTD, suggesting that an elevated postsynaptic spike activity of pyramidal cells enhances GABAergic feedback inhibition and favors LTD induction. The similar form of LTD in the hippocampal CA1, induced by the pairing of postsynaptic hyperpolarization and synaptic activity, has been suggested to be mediated by the activation of mGluRs (Stanton, Chattarji, & Sejnowski, 1991).

Haruta, Kamishita, Hicks, Takahashi, and Tsumoto (1994) have reported that in the rat visual cortical slices, LTD induced by low-frequency stimulation at 1 Hz for 15 min was suppressed by (±)-α-methyl-4-carboxyphenylglycine (MCPG), a selective antagonist for mGluRs. However, LTP induced by θ-burst tetanus for 2 s was not blocked by application of MCPG. A recent study by Stiefel, Tennigkeit, and Singer (2005) demonstrated that the pairing of presynaptic activation with postsynaptic hyperpolarization in layer V pyramidal cells of the rat visual cortex induced reliable LTD, which was dependent on intracellular Ca2+ but not on NMDAR activation.

Based on these experimental findings, we assume that the coincidence of postsynaptic hyperpolarization and mGluR-dependent intracellular [Ca2+] increase in the spine induces LTD, which is likely followed by the collapse of the synaptic connection. The increase in [Ca2+] induced by the release of Ca2+ ions from intracellular calcium stores following the activation of mGluRs in response to the arrival of presynaptic spike activity is expressed as
formula
2.10
Here, the function KmGlu(t) is a kernel of [Ca2+] in response to the activation of mGluRs in the spine. The driving force for LTD induction, is expressed as
formula
2.11
ςHP(t) represents membrane hyperpolarization following depolarization at a cortical neuron. Some form of afterhyperpolarization is generated by intrinsic membrane mechanisms such as the activation of Ca2+- and Na+-dependent K channels (Sanchez-Vives, Nowak, & McCormick, 2000b), which controls the gain of neuronal activity in response to high contrast stimuli (Sanchez-Vives, Nowak, & McCormick, 2000a). Such a gain control requires an increase in the magnitude of hyperpolarization with the stimulus contrast, whereas stimulus-driven neuronal activation, either spike activity or subthreshold depolarization, changes with the stimulus contrast. Therefore, ςHP(t) is phenomenologically modeled by
formula
2.12
Here, Ginh(tt′) describes a kernel that transforms the stimulus-evoked somatic depolarization ζDP of a cortical neuron at t' to the delayed hyperpolarization at t. When equation 2.6 is used, the driving force for LTD induction is given by
formula
2.13
Generally the trivial autocorrelation of a spike train without delay is supposed to be much higher than the autocorrelation with a finite delay or cross-correlation between different presynaptic spike trains. Therefore, when only the contribution of the trivial autocorrelation of a spike train is retained, we obtain
formula
2.14
formula
2.15
where is the average presynaptic spike activity. Equation 2.15 indicates that the coefficient of the LTD driving force is large in the case where postsynaptic hyperpolarization occurs, while an increase of mGluR-mediated [Ca2+] in response to synaptic activation is sustained. For further simplicity, we omit the dependence of the function V0,j on the synaptic position in equation 2.14, replacing it with its spatial average, which should be proportional to the inverse number of total spine sites, . Taken together, the driving force for LTD induction is given by
formula
2.16
In this study, we are interested in the RF formation of a single cortical neuron of an animal reared under a normal visual condition. Thus, we can simply assume that the mean firing rate is uniform over LGN neurons. After all, when a positive constant m is used, the driving force for LTD induction becomes
formula
2.17
where m is the magnitude of synaptic constraint. This driving force provides a synaptic constraint. The more synaptic inputs there are from a single LGN neuron, the more strongly each of the synapses is destabilized.

2.4.  Computer Simulation.

When the local energy at spine j is given by
formula
2.18
the probability of synaptic rewiring is obtained by the following function of the difference between local energies before and after the synaptic update:
formula
2.19
For a sufficiently large β (Nakagama & Tanaka, 2004), afferent inputs are determined so that the system energy is minimized. That is, the maximization of leads to the clustering of correlated afferent inputs on the dendrite of a cortical neuron, whereas the maximization of results in the minimization of the number of afferent inputs from each LGN neuron to the cortical neuron, which prunes exuberant afferent inputs. In this study, we assume for simplicity, that the total number of afferent inputs n0 on the dendrite is constant. Therefore, the maximization of indicates that the number of LGN neurons that send axons to the cortical neuron is maximized. All the values of parameters assumed in the simulation are listed in Table 1 
Table 1:
List of Parameter Values
Cortical CellElectrotonic Length of DendritesλETL = 78 μm
LGN cells Spatial RF Strength ratio of surround and center RFs κ = 1.0 
  RF center extent λc = 0.225 deg 
  RF surround extent λs = 0.9 deg 
 Temporal RF Amplitude of impulse response A = 1.0 
  Time interval of impulse response T0 = 200 ms (ω = 2π/T0
  Delay of response onset τ0 = 57 ms 
  Time constant λ1 = 100 ms 
Grating stimulus  Spatial frequency fs = 0.33 c/deg 
  Temporal frequency ft = 4 Hz 
Cortical CellElectrotonic Length of DendritesλETL = 78 μm
LGN cells Spatial RF Strength ratio of surround and center RFs κ = 1.0 
  RF center extent λc = 0.225 deg 
  RF surround extent λs = 0.9 deg 
 Temporal RF Amplitude of impulse response A = 1.0 
  Time interval of impulse response T0 = 200 ms (ω = 2π/T0
  Delay of response onset τ0 = 57 ms 
  Time constant λ1 = 100 ms 
Grating stimulus  Spatial frequency fs = 0.33 c/deg 
  Temporal frequency ft = 4 Hz 

3.  Results

The synaptic constraint derived from the coincidence mechanism of LTD induction between postsynaptic recurrent inhibition and presynaptic activation is scaled by the number of synapses of afferent inputs, as expressed by equation 2.17. Computer simulation is performed on the basis of the activity-dependent self-organization model combined with the thus-derived synaptic constraint in a many-to-one feedforward network configuration (see Figure 1).

3.1.  Receptive Field Structure Changing with Magnitude of Synaptic Constraint.

First, we show typical examples of self-organizing spatiotemporal RFs, which differ qualitatively for different magnitudes of synaptic constraint (see Figure 2). When the constraint is very weak, spatial RFs are center-surround antagonistic and concentric (see Figure 2A). Either ON- or OFF-center type RFs appear accidentally owing to the initial random arrangements of LGN inputs to a cortical neuron. As observed in Figure 2A, ON- and OFF-responsive subfields are switched from 70 ms to 100 ms. Therefore, all of these RFs are of spatiotemporal separable type. As the constraint becomes stronger, the RF of a cortical neuron becomes larger. The synaptic constraint is imposed for each LGN neuron, which is characteristic of the model presented here. The assumption that any single spine receives an afferent input from the LGN preserves the total number of afferent inputs. Therefore, as the constraint becomes stronger, the cortical neuron tends to receive as many LGN inputs as possible, which gives rise to the expansion of a cortical RF. Reversely, for a very weak synaptic constraint, a few correlated LGN cells send axons to the cortical neuron. Therefore, it is likely that ON- or OFF-center separable RFs are replicated in the cortical neuron with similar structures of LGN cell RFs.

Figure 2:

Typical examples of self-organizing spatiotemporal RFs. (A–F) Spatiotemporal RFs obtained from simulations performed using different magnitudes of synaptic constraint: m= 0.1, 0.8, 1.9, 4.0, 9.0, and 50.0, respectively. The bright and dark domains indicate ON and OFF responsive subfields of cortical neurons, which are composed of inputs from ON- and OFF-center model LGN cells. The brightness and darkness of each domain reflect the strengths of responses of a neuron to ON and OFF stimuli, respectively. The horizontal axes indicate the time step measured from the stimulus onset.

Figure 2:

Typical examples of self-organizing spatiotemporal RFs. (A–F) Spatiotemporal RFs obtained from simulations performed using different magnitudes of synaptic constraint: m= 0.1, 0.8, 1.9, 4.0, 9.0, and 50.0, respectively. The bright and dark domains indicate ON and OFF responsive subfields of cortical neurons, which are composed of inputs from ON- and OFF-center model LGN cells. The brightness and darkness of each domain reflect the strengths of responses of a neuron to ON and OFF stimuli, respectively. The horizontal axes indicate the time step measured from the stimulus onset.

For m ⩾ 0.18, the concentric RF structure is destroyed, and scattered ON- and OFF-responsive subfields emerge (see Figure 2B). Such RFs are also separable in space and time, because ON- and OFF-responsive subfields are switched between 70 ms and 100 ms, as in the case of concentric RFs for m < 0.18.

For m ⩾ 1.34, oriented RFs appear, in which ON- and OFF-responsive subfields are aligned in parallel (see Figure 2C). The angle of this alignment is the preferred orientation of a neuron. As noted in Figure 2C, spatiotemporal RFs remain separable. A further increase in m beyond 2.2 generates spatiotemporal inseparable oriented RFs (see Figure 2D), in which ON- and OFF-responsive subfields move in a direction orthogonal to the axis of subfield elongation.

For m ⩾ 6.9, a cortical neuron exhibits preferred orientations different between early and late responses, showing that the RFs are separable in orientation and time (see Figure 2E). It is likely that the two preferred orientations are at right angles. Because the neuron can represent two orientation preferences, we call such RFs double-oriented RFs. For m ⩾ 23.0, ON- and OFF-responsive subfields are arranged in a checkerboard-like pattern at any time (see Figure 2F), which disrupts orientation preference. Generally the size of RFs increases as the magnitude of synaptic constraint increases, as shown in Figures 2A to 2F, because the constraint tends to increase the number of LGN cells sending inputs to the cortical neuron by reducing the number of inputs from each LGN cell.

Next, we examined the x-t plots for the RFs shown in Figures 2A, 2C, and 2D. Note that ON and OFF responses switch clearly in the time dimension (see Figure 3A1). A similar temporal switch between ON and OFF responses is found in the RF shown in Figure 2C (see Figure 3A2). Therefore, these RFs are clearly classified into the spatiotemporal separable type, as determined by visual inspection of the images of spatiotemporal RFs shown in Figures 2A and 2C. In the x-t plot shown in Figure 3A3, ON and OFF responses are diagonally elongated. This indicates that the RF is inseparable in space and time, as shown in Figure 2D, where ON and OFF subfields move upward. Therefore, it is found that the synaptic constraint can regulate spatiotemporal separability as well as spatial arrangements of ON- and OFF-responsive subfields.

By illustrating polar plots of responses to oriented sinusoidal grating stimuli, we can examine the orientation and direction selectivities of model cortical neurons. For a weak synaptic constraint, the response properties are unselective to stimulus orientation, regardless of whether the subfield arrangement is concentric or scattered (see Figures 3B1 and 3B2). For intermediate magnitudes of synaptic constraint, the responses of the cortical neuron are orientation selective for both the separable and inseparable types (see Figures 3B3 and 3B4). A cortical neuron with an inseparable spatiotemporal RF has a stronger direction selectivity (see Figure 3B4) than a neuron with a separable RF (see Figure 3B3). For stronger synaptic constraints, orientation selectivity is disrupted (see Figures 3B5 and 3B6). Particularly, in the range of 6.9 ⩽ m ⩽ 23.0, a polar plot often shows a four-leaf-type neuron, indicating that the neuron is selective to orthogonal orientations (see Figure 3B5). In an extremely high magnitude of synaptic constraint m>23.0 (see Figure 3B6), the polar plot again shows an unoriented and pandirectional type neuron, similar to the plots shown in Figures 3B1 and 3B2. The response strength of the unoriented neuron shown in Figure 3B6 is one-third those of the unoriented neurons shown in Figures 3B1 and 3B2.

Figure 3:

Typical examples of x-t profiles of cortical neurons (A), polar plots (B), and LGN input patterns (C). A1, A2, and A3 represent the x-t profiles of spatiotemporal RFs shown in Figures 2A, 2C, and 2D, respectively. Numerical indices 1, 2, …, 6 in B and C specify cortical neurons whose spatiotemporal RFs are shown in Figures 2A–2F, respectively. In C, each quadrant panel represents an LGN layer: the upper left, upper right, bottom left, and bottom right panels are layers of the ON-center/nonlagged, ON-center/lagged, OFF-center/nonlagged, and OFF-center/lagged cell types, respectively. The brightness of a pixel indicates the number of inputs from the LGN cell located at a position in the LGN layer.

Figure 3:

Typical examples of x-t profiles of cortical neurons (A), polar plots (B), and LGN input patterns (C). A1, A2, and A3 represent the x-t profiles of spatiotemporal RFs shown in Figures 2A, 2C, and 2D, respectively. Numerical indices 1, 2, …, 6 in B and C specify cortical neurons whose spatiotemporal RFs are shown in Figures 2A–2F, respectively. In C, each quadrant panel represents an LGN layer: the upper left, upper right, bottom left, and bottom right panels are layers of the ON-center/nonlagged, ON-center/lagged, OFF-center/nonlagged, and OFF-center/lagged cell types, respectively. The brightness of a pixel indicates the number of inputs from the LGN cell located at a position in the LGN layer.

Next, we analyze how the RF of a cortical neuron is constructed by LGN inputs to this neuron. For m < 0.18, only a few adjacent LGN cells of the same type send axons to the cortical neuron (see Figure 3C1). As a result, RFs are slightly expanded and become the spatially ON- or OFF-center concentric and temporally lagged or nonlagged types, which are qualitatively the same as LGN RFs. When the magnitude of the constraint is slightly increased, spatial RFs with scattered ON- and OFF-responsive subfields are composed of inputs from ON- and OFF-center LGN cells whose RF centers are mutually separated (see Figure 3C2). In this case, LGN inputs are biased to either the nonlagged or lagged type, indicating that the RF tends to be of the spatiotemporal separable type.

When oriented RFs are formed in the intermediate magnitude of synaptic constraint, the cortical neuron receives inputs from LGN cells whose RF centers are aligned in parallel to one axis in the visual field (see Figures 3C3 and 3C4). Interestingly, LGN inputs are imbalanced between nonlagged and lagged types for weaker constraints (see Figure 3C3), whereas they are likely balanced for stronger constraints (see Figure 3C4). This indicates that a change from the spatiotemporal separable type to the inseparable type occurs in the intermediate range of magnitudes. In double-oriented RFs appearing at larger m, the orientation to which the LGN RF centers are aligned in parallel differs between the nonlagged and lagged types (see Figure 3C5). Consequently, the preferred orientation of the cortical neuron switched temporally, as shown in Figure 2E. At a very large m value, the cortical neuron tends to receive inputs widely from many LGN cells. The RF centers of LGN cells sending inputs to a cortical neuron show a lattice-like arrangement for each LGN cell type (see Figure 3C6). The checkerboard-like subfield arrangement shown in images of the spatiotemporal RF (see Figure 2F) reflects this lattice-like arrangement of LGN inputs.

3.2.  “Phase Transition” of Receptive Field Structure.

We have observed that the RF structure varies with the magnitude of synaptic constraint. Here, we examine how the RF structure changes as the magnitude of synaptic constraint increases or decreases in a quasi-static manner. Starting from a concentric RF at the null synaptic constraint, we gradually increase the magnitude of synaptic constraint m. To analyze structural changes of spatial RFs quantitatively, we calculate the amplitude of the 2D Fourier transform of the spatial RF at each time and plot the maximal Fourier amplitude (MFA) from the response onset to offset against the magnitude of synaptic constraint m in Figures 4A and 4B. Figure 4A shows a single trace of the MFA when m increases (black line) and decreases (gray line), whereas Figure 4B shows superimposed multiple traces when we repeatedly increase and decrease m. Figure 4C shows the changes in LGN RF center maps along the single trace shown in Figure 4A. The upper panels show how the LGN RF center map changes as m increases, and the bottom panels display the map changes as m decreases. At the beginning of the increase in m, the MFA rapidly decreases along the black lines because the number of inputs from each LGN cell to the cortical neuron decreases, whereas a neuron receives more inputs from LGN cells whose RF centers are scattered in the visual field (m = 0.1 to m = 0.8 in the upper panels of Figure 4C). Then, as m increases, the trace changes to a gentle descent in which scattered RFs become oriented RFs with balanced inputs from ON- and OFF-center LGN cells of either the lagged or nonlagged type. Beyond m = 2.2, the trace stops decreasing, showing LGN RF centers aligning in parallel for all LGN cell types (m = 3 in the upper panels of Figure 4C). As m decreases, in turn, the return trace differs from the trace of increasing m and the MFA is larger, because lagged and nonlagged inputs become imbalanced (m = 0.8 in the lower Figure 4C). Consequently, hysteresis is found in the range of 0 ⩽ m ⩽ 3.0. The presence of hysteresis indicates that metastable states exist at the same m. These states are characterized by unoriented RFs (concentric and scattered) and single-oriented RFs (spatiotemporal separable and inseparable) in the range of 0 ⩽ m ⩽ 3.0.

Figure 4:

Traces of the maximal Fourier amplitude (MFA) of spatial RFs when the magnitude of synaptic constraint is changed in a quasi-static manner. (A, D) Single traces of the MFA of spatial RFs averaged over all time steps when the magnitude of synaptic constraint is increased and then decreased. (B, E) Superimposed multiple traces of the MFA from five trials of increase and decrease in the magnitude of synaptic constraint. The black curves indicate the traces when the magnitude of constraint is increased, and the gray curves indicate the traces when the magnitude of constraint is decreased. In A, the vertical dotted lines indicate the magnitudes of constraint of 0.8 and 2.25. (C, F) LGN input patterns along the single traces shown in A and D, respectively. The numerical values shown under some figures indicate the magnitudes of synaptic constraint.

Figure 4:

Traces of the maximal Fourier amplitude (MFA) of spatial RFs when the magnitude of synaptic constraint is changed in a quasi-static manner. (A, D) Single traces of the MFA of spatial RFs averaged over all time steps when the magnitude of synaptic constraint is increased and then decreased. (B, E) Superimposed multiple traces of the MFA from five trials of increase and decrease in the magnitude of synaptic constraint. The black curves indicate the traces when the magnitude of constraint is increased, and the gray curves indicate the traces when the magnitude of constraint is decreased. In A, the vertical dotted lines indicate the magnitudes of constraint of 0.8 and 2.25. (C, F) LGN input patterns along the single traces shown in A and D, respectively. The numerical values shown under some figures indicate the magnitudes of synaptic constraint.

Next, we examine the RF structure changes in the range of 3.0 ⩽ m ⩽ 35.0. Figures 4D and 4E show traces of the MFA as m increases (black lines) or decreases (gray lines) for a single trial and repeated trials, respectively. The traces of MFA for increasing and decreasing m are not clearly distinguishable in the range of 3.0 ⩽ m ⩽ 14.0 (see Figures 4D and 4E). Nevertheless, the spatiotemporal RFs are qualitatively different; Figures 2D and 2E show these RFs to be spatiotemporal inseparable single-oriented and double-oriented, respectively. Moreover, for m = 12, as shown in Figure 4F, RF center maps exhibit parallel alignments of RF centers for all LGN cell types when m is increasing, and the maps exhibit parallel alignments of ON- and OFF-center inputs for the nonlagged type, and 45 degree mismatched alignments of ON- and OFF-center inputs for the lagged type when m is decreasing. These RF center maps indicate that the RF of a cortical neuron is spatiotemporal inseparable single oriented when m is increasing and single oriented at an early response time but unoriented at a late response time when m is decreasing. The MFA degenerates for these two spatiotemporal RF structures, because it is defined as the maximum Fourier amplitude of spatial RFs at either response time.

At 14 ⩽ m ⩽ 20, we can observe hysteresis in Figures 4D and 4E. Accompanied by the emergence of hysteresis, the LGN RF center maps show different structures at the same m (m = 16 in Figure 4F): 45 degree mismatched alignments of ON- and OFF-center inputs for the nonlagged type and parallel alignments for the lagged type when m is increasing, and mismatched alignments for both the lagged and nonlagged types. These RF center maps indicate that RFs are unoriented at an early response time but single oriented at a late response time when m is increasing, and they are unoriented at all response times. The presence of hysteresis indicates the coexistence of metastable states characterized by RF structures.

To further determine the possible distinction among temporal structures of RFs in the range of 3.0 ⩽ m ⩽ 35.0, we calculate the maximal amplitudes of 3D Fourier transforms of spatiotemporal RFs. Figures 5A and 5B show a single trace of the maximal cubic Fourier amplitude (MCFA) and superimposed multiple traces of MCFA, respectively. The different traces for increasing and decreasing m indicate the presence of hysteresis in this range of m. These traces suggest that there are five states of the RF center maps shown in Figure 5C, which correspond to five qualitatively different structures of spatiotemporal RFs. Three of them are stable states characterized by spatiotemporal inseparable single-oriented, double-oriented, and unoriented RFs in some m ranges, which are indicated by the black lines in Figure 5C. The other two types are metastable states in any m range. One is a type of double-oriented RF, composed of lagged and nonlagged inputs whose LGN RF centers are aligned in axes different by 45 degrees; the other metastable state is characterized by early (late) single-oriented and late-(early-) unoriented RFs, as observed at m = 16 in the ascending trace and at m = 12 in the descending trace shown in Figure 4F. These metastable states tend to emerge only when the magnitude of synaptic constraint changes in a quasi-static manner.

Figure 5:

Traces of the maximal 3D Fourier amplitude (MCFA) of spatiotemporal RFs when the magnitude of synaptic constraint is changed in a quasi-static manner. (A) A single trace and (B) superimposed traces of five trials. The conventions on the black and gray curves are the same as those described in Figure 4. (C) A schematic diagram of the MCFAs for stable (black lines) and metastable (gray lines) states. Each state is characterized by LGN RF center alignments shown inside the rectangles. The left and right short line segments inside the rectangles indicate the orientations of LGN RF center alignments for nonlagged and lagged inputs, respectively.

Figure 5:

Traces of the maximal 3D Fourier amplitude (MCFA) of spatiotemporal RFs when the magnitude of synaptic constraint is changed in a quasi-static manner. (A) A single trace and (B) superimposed traces of five trials. The conventions on the black and gray curves are the same as those described in Figure 4. (C) A schematic diagram of the MCFAs for stable (black lines) and metastable (gray lines) states. Each state is characterized by LGN RF center alignments shown inside the rectangles. The left and right short line segments inside the rectangles indicate the orientations of LGN RF center alignments for nonlagged and lagged inputs, respectively.

As for the separability of spatiotemporal RFs at 1.0 ⩽ m ⩽ 4.0, we examined the balance of the relative contributions between nonlagged and lagged inputs to a cortical neuron, using the balance index (BI) defined by
formula
3.1
where Nnonlagged and Nlagged are the numbers of inputs from nonlagged and lagged LGN cells, respectively. The balance index (BI) is plotted in the black solid curve in Figure 6. The BI increases nearly monotonically from 0 to 1 as m increases from 0 to 3. We also calculate the BI for ON- and OFF-center inputs instead of lagged and nonlagged inputs, which is shown as a gray curve in Figure 6. The relative contribution between ON- and OFF-center inputs becomes rapidly balanced as m increases from 0 and reaches a saturation level at approximately m = 1.5. The behavior of the BI indicates that although cortical neurons are likely to have balanced ON and OFF subfields even at weak constraints, the integration of inputs from lagged and nonlagged LGN cells requires stronger constraints. Therefore, at weaker constraints, a cortical neuron tends to exhibit a spatiotemporal separable RF that is composed of either nonlagged or lagged inputs. At stronger constraints, it is quite likely that RFs are inseparable in space and time, which needs balanced inputs from lagged and nonlagged LGN cells.
Figure 6:

Balance of relative contributions between nonlagged and lagged inputs or ON- and OFF-center inputs depending on the magnitude of synaptic constraint. The balance indices (BI; definition is given in the text) are plotted against the magnitude of synaptic constraint for the lagged and nonlagged input balance (black curve) and for the ON- and OFF-center input balance (gray curve). BI = 1 indicates that a cortical neuron receives the same number of inputs from opposite types of LGN cell, whereas BI = 0 indicates that a neuron receives inputs from either type of LGN cell.

Figure 6:

Balance of relative contributions between nonlagged and lagged inputs or ON- and OFF-center inputs depending on the magnitude of synaptic constraint. The balance indices (BI; definition is given in the text) are plotted against the magnitude of synaptic constraint for the lagged and nonlagged input balance (black curve) and for the ON- and OFF-center input balance (gray curve). BI = 1 indicates that a cortical neuron receives the same number of inputs from opposite types of LGN cell, whereas BI = 0 indicates that a neuron receives inputs from either type of LGN cell.

Finally, we examined the equilibrium distribution of the self-organizing spatiotemporal RFs obtained from a long-run simulation at each magnitude of synaptic constraint. Such a long-run simulation minimizes the possibility of the emergence of metastable states, which are found in quasi-static changes in the magnitude of synaptic constraint. The scattered plot of the MCFA and its standard deviation against the magnitudes of synaptic constraint are shown in Figure 7 on the logarithmic scale. At the magnitudes of constraint that cause the transitions of RF structures, the standard deviation increases. Consequently, we can identify six phases corresponding to different spatiotemporal RF structures for the wide-magnitude range of synaptic constraint. Phases a, b, c, d, e, and f, respectively, correspond to the following RF structures: concentric unoriented RFs, subfield-scattered RFs, single-oriented separable RFs, single-oriented inseparable RFs, double-oriented RFs, and checkerboard-like unoriented RFs.

Figure 7:

Scatter plot of maximal amplitudes of 3D Fourier transforms of spatiotemporal RFs (top) and the standard deviation of maximal amplitudes (bottom) against the magnitude of synaptic constraint. The standard deviation of maximal Fourier amplitudes is maximal at the magnitude of synaptic constraint at which the maximal amplitudes scatter widely owing to the presence of hysteresis. There are five major peaks in B, which indicate transitions through six successive phases (a–f), in which qualitatively different structures of spatiotemporal RFs appear. In phases a, b, and c, concentric RFs, scattered RFs, and single-oriented RFs appear, all of which are of the spatiotemporal separable type. In phase d, spatiotemporal inseparable single-oriented RFs appear. In phase e, double-oriented RFs appear. Finally in phase e, unoriented checkerboard-like RFs appear.

Figure 7:

Scatter plot of maximal amplitudes of 3D Fourier transforms of spatiotemporal RFs (top) and the standard deviation of maximal amplitudes (bottom) against the magnitude of synaptic constraint. The standard deviation of maximal Fourier amplitudes is maximal at the magnitude of synaptic constraint at which the maximal amplitudes scatter widely owing to the presence of hysteresis. There are five major peaks in B, which indicate transitions through six successive phases (a–f), in which qualitatively different structures of spatiotemporal RFs appear. In phases a, b, and c, concentric RFs, scattered RFs, and single-oriented RFs appear, all of which are of the spatiotemporal separable type. In phase d, spatiotemporal inseparable single-oriented RFs appear. In phase e, double-oriented RFs appear. Finally in phase e, unoriented checkerboard-like RFs appear.

4.  Discussion

4.1.  Comparisons of RFs in Primary Visual Cortex.

There are different types of RF of neurons in the geniculate-recipient layer of the primary visual cortex in different species. In layer 4Cβ of the primary visual cortex of macaques, which receives afferent inputs from the parvocellular layers of the LGN, all neurons have ON- or OFF-center concentric RFs (Blasdel & Fitzpatrick, 1984). This RF structure agrees well with the simulated RF structure for a very weak synaptic constraint in our model. In layer 4 of area 17 of cats, neurons are classified into simple cells whose RFs are composed of ON- or OFF-responsive subfields elongated along the axes parallel to the preferred orientations of the neurons (Hirsch et al., 1998). Among these simple cells are spatiotemporal separable and inseparable types, which correspond to direction-nonselective and direction-selective simple cells (DeAngelis, Ohzawa, & Freeman, 1995). Both types of RF are reproduced by our simulations at different magnitudes of synaptic constraint. Saul and Humphrey (1992) reported three types of simple cell—nonlagged-like, lagged-like and mixed types—in the geniculate-recipient layer of the primary visual cortex. These nonlagged- and lagged-like cells perhaps correspond to model neurons exhibiting spatiotemporal separable single-oriented RFs at m < 2.2. The mixed type of simple cell likely corresponds to neurons with spatiotemporal inseparable single-oriented RFs at m>2.2. Furthermore, in the cat primary visual cortex, not only single-oriented RFs but also double-oriented RFs have been found to exist (Worgotter & Eysel, 1991; Shevelev et al., 1994). Our simulations also predict the emergence of such double-oriented RFs. To our knowledge, the double-oriented RFs are derived from a mathematical model for the first time. Taken together, this study suggests that only a single parameter, the magnitude of synaptic constraint, can regulate wiring patterns of afferent inputs during development, thereby resulting in a repertoire of spatiotemporal RFs found in previous experiments.

4.2.  Interpretation of Other Experiments on Visual Cortex.

It has been demonstrated that in the primary visual cortex of kittens reared under monocular deprivation concomitant with muscimol infusion, the ocular dominance of neuronal responses paradoxically shifted to the deprived eye, as opposed to the normal shift to the nondeprived eye (Reitor & Stryker, 1988). This physiological phenomenon was supported by the anatomical observation in the same protocol, which showed that the cortical territory occupied by geniculocortical afferents from the nondeprived eye became smaller, whereas that from the deprived eye became larger (Hata & Stryker, 1994). This paradoxical ocular dominance shift may be accounted for by our model. When muscimol is infused in the cortex, neurons are hyperporlarized and do not elicit spikes. Therefore, the NMDAR-dependent Hebbian term given by equation 2.9 should be ineffective for all cortical spines. The membrane potential of cortical neurons is tonically hyperpolarized, indicating that ςHP in equation 2.11 is a negative constant. The mGluR-mediated intracellular [Ca2+] rise occurs only at spines that receive nondeprived-eye inputs and is proportional to the mean firing rate. Consequently, synaptic constraint working at spines receiving nondeprived-eye inputs induces LTD at synapses, indicating that ocular dominance is shifted to the deprived eye. If the LTD finally leads to synapse elimination, deprived-eye inputs predominantly occupy the visual cortex, as observed by Hata and Stryker (1994). With respect to this successful explanation of the muscimol-induced reverse ocular dominance shift, the model we present also seems biologically plausible.

The synaptic plasticity modeled in this study is based on a push-pull mechanism derived from NMDAR-mediated LTP (Hebbian learning) and mGluR-mediated LTD (synaptic constraint), which suggests that different spatiotemporal patterns of [Ca2+] increase are important. This may be justified by the idea that intracellular [Ca2+] elevation has multiple functions depending on the channels through which calcium ions are taken up due to the difference in their subcellular localization and temporal properties (Furuyashiki, Arakawa, Takemoto-Kimura, Bito, & Narumiya, 2002).

4.3.  Interpretation of LGN Neurons' RFs.

It is also believed that RFs of LGN cells are concentric, center-surround antagonistic similarly to those of retinal ganglion cells in monkeys (Wiesel & Hubel, 1966) and cats (Hubel & Wiesel, 1961; DeAngelis et al., 1995), although several reports suggested that a portion of LGN relay cells are orientation biased (Vidyasagar & Urbas, 1982; Thompson, Leventhal, Zhou, & Liu, 1994; Xu, Ichida, Shostak, Bonds, & Casagrande, 2002). Despite the observation that LGN receives inputs from both ON- and OFF-center retinal ganglion cells, which may induce synaptic competition, why do LGN cells tend to be unoriented? This can also be explained by an extremely weak synaptic constraint on afferent retinal inputs during the formation of retinogeniculate connections: A weak constraint allows retinal inputs from a few single retinal ganglion cells of the same type to innervate an LGN cell. Such a connectivity pattern is consistent with the experimental observation of connections from retinal ganglion cells to LGN relay cells (Hubel & Wiesel, 1961; Dubin & Cleland, 1977), which allows LGN cells to succeed in forming a concentric RF organization of retinal ganglion cells (Shapley & Lennie, 1985). Although a recent cross-correlation analysis indicates that several retinal inputs converge into a single LGN cell, only inputs from retinal ganglion cells sharing precisely similar response properties are terminated at an LGN cell (Hirsch et al., 1998). This connection pattern is supported by a linear mechanism analysis performed by Soodak, Shapley, and Kaplan (1987) and later confirmed experimentally by Smith, Chino, Ridder, Kitagawa, and Langston (1990). These studies suggest that even in the presence of an orientation bias in LGN cells, the bias primarily reflects the functional properties of individual retinal ganglion cells. Nevertheless, if there are LGN cells with dendritic arbors sufficiently larger than the electrotonic distance, we cannot use the approximation of replacing V0,j with its average value in equation 2.16. This indicates that the synaptic constraint becomes effectively weaker, and hence the LGN cells may have orientation selectivity. The previous reports on the orientation-biased cells in the LGN (Vidyasagar & Urbas, 1982; Thompson et al., 1994; Xu et al., 2002) may partly support the existence of such neurons with large dendritic arbors. Accordingly, our model predicts that the orientation selectivity of LGN cells depends on the size of their dendritic arbors.

4.4.  Comparison with Synaptic Constraints in Other Models.

The contribution of some form of constraint on synaptic connections to the self-organization of afferent inputs is indispensable for the formation of biologically plausible response properties of postsynaptic neurons. Since von der Malsburg's pioneering work (1973), in many self-organization models, a constant sum of synaptic strengths over all presynaptic cells has been assumed (Swindale, 1996).

Miller and Mackay (1994) analyzed in detail how the synaptic constraint regulates self-organizing afferent inputs and resulting RFs. They compared subtractive and multiplicative normalization conditions with an unconstrained condition for RF formation. In our model, these types of synaptic constraint are originally included as the basic properties of synaptic input variables that have been derived from a winner-take-all mechanism of presynaptic inputs competing for a postsynaptic dendritic site (Tanaka, 1990). Our synaptic constraint is related to the upper limit of synaptic strength imposed on the synaptic strength of each afferent input, which is implicitly assumed in the previous model (Miller & Mackay, 1994), because a high magnitude of synaptic constraint suppresses the number of synaptic inputs from each presynaptic neuron. Miller and Mackay (1994) showed that when the upper limit of the synaptic strength of inputs from a single presynaptic neuron is assumed to be half of the constant sum of the synaptic strengths in a subtractive normalization condition, oriented RFs appear. This may correspond to the case of intermediate magnitude of synaptic constraint in our model, where we obtain oriented RFs.

Oshiro and Weliky (2006) demonstrated that “split synaptic constraint,” which normalizes the total synaptic inputs from ON- and OFF-center LGN cells separately, works well to reproduce simple-cell-like RFs under the assumption of monotonically decreasing (simple falloff) presynaptic correlation functions, as observed electrophysiologically between pairs of LGN cells in immature ferrets. This synaptic constraint may require molecular markers that identify ON- and OFF-center inputs separately. However, no such molecular markers have been found so far. Therefore, the split synaptic constraint seems to be unlikely. When the magnitude of synaptic constraint proposed here is in the intermediate range in which oriented RFs appear, ON- and OFF-center inputs are balanced. In this magnitude range, our synaptic constraint may work similarly to the split synaptic constraint. In this sense, the split synaptic constraint is regarded as a special case of our synaptic constraint.

4.5.  Limitation and Perspectives of the Model.

We propose here a form of synaptic constraint on geniculocortical afferent inputs to a single visual cortical neuron based on a mechanism underlying heterosynaptic LTD, which is induced by the coincidence of presynaptic activation and postsynaptic hyperpolarization induced by local recurrent inhibition. This synaptic constraint can regulate the qualitative structure of spatiotemporal RFs of cortical neurons via phase transition phenomena. The entire repertoire of RFs of neurons in the geniculate-recipient layer of the primary visual cortex of different species is reproduced by the control of a single parameter, that is, the magnitude of synaptic constraint. On the other hand, our synaptic constraint derived from an mGluR function works to preserve a normal ocular dominance distribution against monocular deprivation. Most experiments on mGluR functions in the context of ocular dominance plasticity have attempted to examine homosynaptic LTD, but not local-circuit-mediated heterosynaptic LTD. It has been demonstrated that homosynaptic LTD induced by the activation of mGluR2 is not relevant to the ocular dominance shift in kittens (Hensch & Stryker, 1996). If the ocular dominance shift is determined by the push-pull mechanism of NMDAR-mediated LTP and mGluR-mediated LTD, as assumed in the model, the blockade of either mechanism may not be able to stop the shift.

Also, our model describes only the RF formation of a single geniculate-recipient neuron in the visual cortex. We did not discuss circuit effects because we neglected cortical lateral interaction. It is, however, considered that a lateral interaction may work to form clustering of neurons of similar preferences regarding the orientation and direction of motion, that is, the orientation and direction maps. If this is the case, our model is expected to explain RF formation sufficiently. The justification of our model remains as a future experimental research target.

Appendix:  Receptive Field Profiles of LGN Cells and Responses to Visual Stimuli

The RFs of LGN cells are classified into the spatiotemporal separable type (DeAngelis et al., 1995), which is represented by the product of the spatial component and the temporal component ,
formula
A.1
k and l represent the position of an LGN cell and the visual field location, respectively. Because we assume that all LGN cells are arranged retinotopically, k also represents the location of the RF center of an LGN cell in the visual field. μ1 = +1 or μ1 = −1 denotes an ON- or OFF-center type input, and μ2 = +1 or μ2 = −1 denotes a nonlagged or lagged type input. The spatial component of an LGN RF is given by the difference of gaussian (DOG) function,
formula
A.2
which is a function of the distance dk,l between the position of the RF center k of an LGN cell and a given position l in the visual field. λc and λs indicate the extent of the center excitatory subfield and surround suppressive subfield of the spatial component of the RF, respectively. κ is the ratio of the total response strength of center excitation to surround suppression. The ON- (OFF-) center nonlagged cells exhibit early excitation and late suppression in response to a light (dark) spot in the center subfield, whereas the ON- (OFF-) center lagged cells show early suppression and late excitation in response to a light (dark) spot in the center subfield (DeAngelis et al., 1995). To reproduce these properties, we employ the following formula as temporal components of RFs of LGN cells:
formula
A.3
Here A and ω are the impulse response strength of the LGN cell and the temporal frequency, respectively. T0, τ0, and λt are the time interval during which LGN cells respond to a stimulus impulse, the delay of a response onset, and the time constant of response attenuation, respectively. The values of these parameters were determined so that typical spatiotemporal RFs measured by the reverse correlation method in the electrophysiological recordings (DeAngelis et al., 1995) were best fitted.
The firing rate of an LGN cell at time t, , in response to the temporally changing contrast of visual stimulus zl(t) presented in the visual field, is written as
formula
A.4
The rectification function F(x) defined by F(x) = max(x, 0) represents the transformation of membrane potential to firing rate. In this study, the visual stimuli that we presented in the model visual field were sinusoidal gratings drifting in the direction θDR with a spatial frequency fs and a temporal frequency ft, which are given by
formula
A.5

References

Bear
,
M. F.
,
Kleinschmidt
,
A.
,
Gu
,
Q. A.
, &
Singer
,
W.
(
1990
).
Disruption of experience-dependent synaptic modifications in striate cortex by infusion of an NMDA receptor antagonist
.
J. Neurosci.
,
10
,
909
925
.
Blasdel
,
G. G.
, &
Fitzpatrick
,
D.
(
1984
).
Physiological organization of layer 4 in macaque striate cortex
.
J. Neurosci.
,
4
,
880
895
.
DeAngelis
,
G. C.
,
Ohzawa
,
I.
, &
Freeman
,
R. D.
(
1995
).
Receptive-field dynamics in the central visual pathways
.
Trans in Neurosci.
,
18
,
451
458
.
Dubin
,
M. W.
, &
Cleland
,
B. G.
(
1977
).
Organization of visual inputs to interneurons of lateral geniculate nucleus of the cat
.
J. Neurophysiol.
,
40
,
410
427
.
Furuyashiki
,
T.
,
Arakawa
,
Y.
,
Takemoto-Kimura
,
S.
,
Bito
,
H.
, &
Narumiya
,
S.
(
2002
).
Multiple spatiotemporal modes of actin reorganization by NMDA receptors and voltage-gated Ca2+channels
.
Proc. Nat. Academy Science
,
99
,
14458
14463
.
Gilbert
,
C. D.
, &
Wiesel
,
T. N.
(
1979
).
Morphology and intracortical projections of functionally characterized neurons in the cat visual cortex
.
Nature
,
280
,
120
125
.
Gu
,
Q. A.
,
Bear
,
M. F.
, &
Singer
,
W.
(
1989
).
Blockade of NMDA-receptors prevents ocularity changes in kitten visual cortex after reversed monocular deprivation
.
Dev. Brain Res.
,
47
,
281
288
.
Haruta
,
H.
,
Kamishita
,
T.
,
Hicks
,
T. P.
,
Takahashi
,
M. P.
, &
Tsumoto
,
T.
(
1994
).
Induction of LTD but not LTP through metabotropic glutamate receptors in visual cortex
.
NeuroReport
,
5
,
1829
1832
.
Hata
,
Y.
, &
Stryker
,
M. P.
(
1994
).
Control of thalamocortical afferent rearrangement by postsynaptic activity in developing visual cortex
.
Science
,
265
,
1732
1735
.
Hensch
,
T. K.
, &
Stryker
,
M. P.
(
1996
).
Ocular dominance plasticity under metabotropic glutamate receptor blockade
.
Science
,
272
,
554
557
.
Hirsch
,
J. A.
,
Alonso
,
J.-M.
,
Reid
,
R. C.
, &
Martinez
,
L. M.
(
1998
).
Synaptic integration in striate cortical simple cells
.
J. Neurosci.
,
18
,
9517
9528
.
Hubel
,
D. H.
, &
Wiesel
,
T. N.
(
1961
).
Integrative action in the cat's lateral geniculate body
.
J. Physiol.
,
155
,
385
398
.
Humphrey
,
A. I.
, &
Norton
,
T. T.
(
1980
).
Topographic organization of the orientation column system in the striate cortex of the tree shrew (Tupaia gilis). I. Microelectrode recording
.
J. Comp. Neurol.
,
192
,
531
547
.
Kleinschmidt
,
A.
,
Bear
,
M. F.
, &
Singer
,
W.
(
1987
).
Blockade of “NMDA” receptors disrupts experience-dependent plasticity of kitten striate cortex
.
Science
,
238
,
355
358
.
Linsker
,
R.
(
1986
).
From basic network principles to neural architecture: Emergence of spatial-opponent cells
.
Proc. Natl. Acad. Sci. USA
,
83
,
7508
7512
.
Martin
,
K. A. C.
, &
Whitteridge
,
D.
(
1984
).
The relationship of receptive field properties to the dendritic shape of neurons in the cat striate cortex
.
J. Physiol.
,
356
,
291
302
.
McConnel
,
S. K.
, &
LeVay
,
S.
(
1986
).
Anatomical organization of the visual system of the mink
.
Mustela vision. J. Comp. Neurology
,
250
,
109
132
.
Metropolis
,
N.
,
Rosenbluth
,
A. W.
,
Rosenbluth
,
M. N.
,
Teller
,
A. H.
, &
Teller
,
E.
(
1953
).
Equation of state calculations by fast computing machines
.
J. Chem. Phys.
,
21
,
1087
1092
.
Miller
,
K. D.
(
1994
).
A model for the development of simple cell receptive fields and the ordered arrangement of orientation columns through the activity dependent competition between ON- and OFF-center inputs
.
J. Neurosci.
,
14
,
409
441
.
Miller
,
K. D.
,
Keller
,
J. B.
, &
Stryker
,
M. P.
(
1989
).
Ocular dominance column development: Analysis and simulation
.
Science
,
245
,
605
615
.
Miller
,
K. D.
, &
Mackay
,
D. J.
(
1994
).
The roles of constraints in Hebbian learning
.
Neural Computation
,
6
,
100
126
.
Nakagama
,
H.
, &
Tanaka
,
S.
(
2004
).
Self-organization model of cytochrome oxidase blobs and ocular dominance columns in the primary visual cortex
.
Cereb. Cortex
,
14
,
376
386
.
Nevian
,
T.
, &
Sakmann
,
B.
(
2004
).
Single spine Ca2+signals evoked by coincident EPSPs and backpropagating action potentials in spiny stellate cells of layer 4 in the juvenile rat somatosensory barrel cortex
.
J. Neurosci.
,
24
,
1689
1699
.
Ohshiro
,
T.
, &
Weliky
,
M.
(
2006
).
Simple fall-off pattern of correlated neural activity in the developing lateral geniculate nucleus
.
Nat. Neurosci.
,
9
,
1541
1548
.
Ramoa
,
A. S.
,
Paradiso
,
M. A.
, &
Freeman
,
R. D.
(
1988
).
Blockade of intracortical inhibition in kitten striate cortex: Effects on receptive field properties and associated loss of ocular dominance plasticity
.
Exp. Brain Res.
,
73
,
285
296
.
Reitor
,
H. O.
, &
Stryker
,
M. P.
(
1988
).
Neural plasticity without postsynaptic action potentials: Less-active inputs become dominant when kitten visual cortical cells are pharmacologically inhibited
.
Proc. Natl. Acad. Sci. U.S.A.
,
85
,
3623
3627
.
Sanchez-Vives
,
M. V.
,
Nowak
,
L. G.
, &
McCormick
,
D. A.
(
2000a
).
Membrane mechanisms underlying contrast adaptation in cat area 17 in vivo
.
J. Neurosci.
,
20
,
4267
4285
.
Sanchez-Vives
,
M. V.
,
Nowak
,
L. G.
, &
McCormick
,
D. A.
(
2000b
).
Cellular mechanisms of long-lasting adaptation in visual cortical neurons in vitro
.
J. Neurosci.
,
20
,
4286
4299
.
Saul
,
A. B.
, &
Humphrey
,
A. L.
(
1992
).
Evidence of input from lagged cells in the lateral geniculate nucleus to simple cells in cortical area 17 of the cat
.
J. Neurophysiol.
,
68
,
1190
1208
.
Shapley
,
R.
, &
Lennie
,
P.
(
1985
).
Spatial frequency analysis in the visual system
.
Ann. Rev. Neurosci.
,
8
547
583
.
Shevelev
,
I. A.
,
Lazareva
,
N. A.
,
Novikova
,
B. V.
,
Tikhomirov
,
A. S.
, &
Sharaev
,
G. A.
(
1994
).
Double orientation tuning in the cat visual cortex units
.
Neurosci.
,
61
,
965
973
.
Smith
,
E. L. III
,
Chino
,
Y. M.
,
Ridder
,
W. H. III
,
Kitagawa
,
K.
, &
Langston
,
A.
(
1990
).
Orientation bias of neurons in the lateral geniculate nucleus of macaque monkeys
.
Vis. Neurosci.
,
5
,
525
545
.
Soodak
,
R. E.
,
Shapley
,
R. M.
, &
Kaplan
,
E.
(
1987
).
Linear mechanism of orientation tuning in the retina and lateral geniculate nucleus of the cat
.
J. Neurophysiol.
,
58
,
267
275
.
Stanton
,
P. K.
,
Chattarji
,
S.
, &
Sejnowski
,
T. J.
(
1991
).
2-amino-3-phosphonopropionic acid, an inhibitor of glutamate-stimulated phosphoinositide turnover, blocks induction of homosynaptic long-term depression, but not potentiation, in rat hippocampus
.
Neurosci. Lett.
,
127
,
61
66
.
Steel
,
P. M.
, &
Mauk
,
M. D.
(
1999
).
Inhibitory control of LTP and LTD: Stability of synapse strength
.
J. Neurophysiol.
,
81
,
1559
1566
.
Stiefel
,
K. M.
,
Tennigkeit
,
F.
, &
Singer
,
W.
(
2005
).
Synaptic plasticity in the absence of backpropagating spikes of layer II inputs to layer V pyramidal cells in rat visual cortex
.
Eur. J. Neurosci.
,
21
,
2605
2610
.
Swindale
,
N. V.
(
1996
).
The development of topography in the visual cortex: A review of models
.
Network
,
7
,
161
247
.
Tanaka
,
S.
(
1990
).
Theory of self-organization of cortical maps: Mathematical framework
.
Neural Networks
,
3
,
625
640
.
Thiels
,
E.
,
Barrionuevo
,
G.
, &
Berger
,
T. W.
(
1994
).
Excitatory stimulation during postsynaptic inhibition induces long-term depression in hippocampus in vivo
.
J. Neurophysiol.
,
72
,
3009
3016
.
Thomas
,
P. J.
, &
Cowan
,
J. D.
(
2006
).
Simultaneous constraints on pre- and post-synaptic cells couple cortical feature maps in a 2D geometric model of orientation preference
.
Math. Med. Biol.
,
23
,
119
138
.
Thompson
,
K. G.
,
Leventhal
,
A. G.
,
Zhou
,
Y.
, &
Liu
,
D.
(
1994
).
Stimulus dependence of orientation and direction sensitivity of cat LGNd relay cells without cortical inputs: A comparison with area 17
.
Visual Neurosci.
,
11
,
939
951
.
Vidyasagar
,
T. R.
, &
Urbas
,
J. V.
(
1982
).
Orientation sensitivity of cat LGN neurons with and without inputs from visual cortical areas 17 and 18
.
Exp. Brain Res.
,
46
,
157
169
.
von der Malsburg
,
C.
(
1973
).
Self-organization of orientation sensitive cells in the striate cortex
.
Kybernetik
,
14
,
85
100
.
Wiesel
,
T. N.
, &
Hubel
,
D. H.
(
1966
).
Spatial and chromatic interactions in the lateral geniculate body of the rhesus monkey
.
J. Neurophysiol.
,
29
,
1115
1156
.
Wilson
,
J. R.
, &
Sherman
,
S. M.
(
1976
).
Receptive-field characteristics of neurons in cat striate cortex: Changes with visual field eccentricity
.
J. Neurophysiol.
,
39
,
512
532
.
Worgotter
,
F.
, &
Eysel
,
U. T.
(
1991
).
Axial responses in visual cortical cells: Spatio-temporal mechanisms quantifies by Fourier components of cortical tuning curves
.
Exp. Brain Res.
,
83
,
656
664
.
Xu
,
X.
,
Ichida
,
J.
,
Shostak
,
Y.
,
Bonds
,
A. B.
, &
Casagrande
,
V. A.
(
2002
).
Are primate lateral geniculate nucleus (LGN) cells really sensitive to orientation and direction?
Vis. Neurosci.
,
19
,
97
108
.
Yang
,
X. D.
,
Connor
,
J. A.
, &
Faber
,
D. S.
(
1994
).
Weak excitation and simultaneous inhibition induce long-term depression in hippocampal CA1 neurons
.
J. Neurophysiol.
,
71
,
1586
1590
.