Stabilized Supralinear Network Model of Responses to Surround Stimuli in Primary Visual Cortex

A, Density of retrogradely labeled excitatory cells in macaque V1 vs. distance from the retrograde tracer injection site in V1. This is proportional to the probability that a neuron at one site in V1 projects to another V1 site, as a function of the distance between them. The line is an exponential fit to the data, with a length constant of 230 μm. Reproduced with permission from Markov et al. (2011). B. For the connection strengths used in the model, we plot the average strength of a weight between two cells a given distance apart, averaged over all cell pairs, for a connection from an excitatory cell to an excitatory cell (W_EE, left) or to an inhibitory cell (W_IE, right). This reflects both the explicit dependence on distance, p_XE(|x_a − x_b|), and the dependence q_XE(|θ_a − θ_b|) on preferred orientation difference along with the orientation map (Fig. 1A) (the functions p and q are defined in the text). The bump in the right panel arises because E → I long range connections have a Gaussian spatial profile that remains nonzero around 1 mm, where the neuron’s preferred orientation tends to recur since this is roughly the period of the orientation map.

SSN behavior of the model network A, Inputs to an excitatory (E) cell and its firing rate vs external input. The cell is at a randomly selected grid location (see Section Model, subsection Model details). Stimulus contrast level corresponding to external input is shown on the bottom axis. The net input is defined as (I_ext + I_{exc−recurrent} − I_inh), where I_ext is the external input to the cell, I_{exc−recurrent} is the cell’s recurrent excitatory input from the network, and I_inh its recurrent inhibitory input from the network, I_inh is defined to be positive, see Section Model, subsection Model details for the expressions of I_{exc−recurrent} and I_inh. B, The network is dominated by external input for weaker stimuli (weaker external input) and by network (recurrent) inputs for stronger stimuli. Plot shows percentage of external and network inputs as a function of external input for the excitatory cell in (A) and an inhibitory cell at the same grid location (dashed line is external input, solid line is network input). Here, the total input is defined as (I_ext + I_{exc−recurrent} + I_inh), and the network input is (I_{exc−recurrent} + I_inh). C, Network input is increasingly inhibition-dominated with increasing stimulus strength. Plot shows percentage of network input that is excitatory, I_{exc−recurrent}/(I_{exc−recurrent} + I_inh), as a function of external input for the excitatory and inhibitory cells in (A, B). In panels (A–C) we use a stimulus of width 2.16° centered on the cell’s retinotopic position and with the cell’s preferred orientation. D, Surround Facilitation to Suppression transition: a near surround can be facilitating or suppressing depending on the center stimulus contrast. SI_m negative means facilitation, while SI_m positive means suppression (see Section Model, subsection Model details, Eq. 10 for the definition of SI_m). In panel (D), the data are from an excitatory cell at a randomly selected grid location (see Section Model, subsection Model details); surround stimulus has contrast C = 12, and inner and outer widths 0.865° and 4.32° respectively; the center stimulus width is 0.65° and its contrast is shown on the x-axis; both center and surround stimuli are centered on the cell’s retinotopic position, with the cell’s preferred orientation. The inputs are in arbitrary units (a.u.).

Surround suppression (A, B) The firing rates of an excitatory (E) cell (A) and an inhibitory (I) cell at the same grid location (B) vs. stimulus size. Different colors correspond to different stimulus contrast levels, high (C = 16.4; external input 80% of maximal), medium (C = 10; external input 42% of maximal) and low (C = 8; external input 25% of maximal). The cells are at a randomly selected grid location (see Section Model, subsection Model details). C, The average firing rate of 80 E cells at randomly selected grid locations (see Section Model, subsection Model details), and of 80 I cells at the same grid locations, after normalizing each cell’s rates so that its peak rate is 1.0, vs. stimulus size for a high contrast stimulus (C = 16.4). D, The distribution of Summation Field sizes (SFS) of the E and I cells used to produce panel (C), the mean SFS for the E cells is 1.14 deg and for the I cells is 1.75 deg. E, The mean suppression index of the E cells and I cells used to produce panel (C), versus stimulus contrast. The mean Suppression Index (SI) for E and I cells changes from little or no suppression (low SI’s) for very weak stimuli to stronger suppression (higher SI’s) for stronger stimuli, with E cells showing much stronger suppression than I cells. The error bars are of order 10⁻² or smaller.

Surround suppression, summation field sizes A, The distribution of summation field sizes for 80 excitatory (E) cells (same E cells used to produce panel Fig. 4C), at contrast C = 16.4 (dark red color) and contrast C = 10 (light red color). B, The distribution of the ratio of the summation field sizes in (A). The summation field size of all cells is smaller for the higher contrast stimulus.

Surround suppression, inputs to cells A, The excitatory (red) and inhibitory (blue) total input to the excitatory (E) cell in Figure 4A, shown for the high contrast stimulus (C = 16.4; external input 80% of maximal), both show surround suppression. (B, C) The size tuning of the averaged normalized excitatory and inhibitory inputs (each normalized to have peak value 1) to excitatory (E) cells (B) and inhibitory (I) cells (C) for contrast C = 16.4 (same cells used to produce panel Fig. 4C). Note the change in horizontal axis between panels (A) and (B, C). D, The excitatory and inhibitory inputs to E cells (same E cells used to produce panel Fig. 4C) for a large stimulus (for which all the cells are surround suppressed) are shown vs. their values for a small stimulus (with size given by the average size that yields maximal response across cells, see Section Results, subsection Surround suppression). Panel (E) is the same as (D) but for I cells (same I cells used to produce panel Fig. 4C). Stimulus contrast C = 16.4. The inputs are in arbitrary units (a.u.).

Surround tuning to the center orientation in an excitatory (E) cell The orientations are relative to the cell’s preferred orientation. The black curves show the cell’s orientation tuning curve for a center-only stimulus (i.e., firing rate vs. center orientation), normalized so the maximum response is 1.0. The red curves show the similarly normalized tuning to surround orientation for a fixed center stimulus. In each panel, the red asterisk marks the fixed center orientation: A, center at preferred minus 20°, (B) center at preferred and (C) center at preferred plus 20°. This cell is at a randomly sampled grid location (see Section Model, subsection Model details).

Surround tuning to the center orientation Surround tuning in 67 excitatory (E) cells at randomly selected grid locations (see Section Model, subsection Model details). Both center and surround orientations are varied from preferred minus 40° to preferred plus 40° in increments of 10°. A, Average surround modulation map. For each cell, the map is obtained by dividing center-surround responses by the corresponding center-only responses, each row has its minimum value subtracted. B, Histogram of the difference between the orientation of the surround that maximally suppresses the cell’s firing rate, and the center stimulus orientation. The data is pooled over all cells and center orientations. C, Whisker plot of orientation of the surround that maximally suppresses the cell’s firing rate against the center stimulus orientation, the box extends from quantile Q1 to Q3, the orange line is the median. The upper whisker extends to last datum less than Q3 + k * IQR, similarly, the lower whisker extends to the first datum greater than Q1 − k * IQR, where IQR is the interquartile range (Q3-Q1) and k = 1.5, the circles represent the outlier data. In (A) and (C) orientations are shown relative to preferred.

Feature-specific surround suppression A, The firing rate of a small population of neurons in response to a center stimulus. The neurons are binned in 5° bins according to their preferred orientation. The dots are the data points, and the lines are the von-Mises-function fits to the data. The medium gray points and dashed line are the population response to the first component of the plaid (P₁). The light gray points and dashed line are the population response to the second component of the plaid (P₂). The dark gray points and solid line are the population response to the plaid. The black points and solid line are the population response to the plaid in the presence of a surround stimulus whose orientation matches the plaid’s second component. B, C, values of w₁ and w₂ (the weightings in fitting the plaid population response to a weighted sum of the two component responses), for a 60° center plaid stimulus, shown for 12 different plaid rotations (every 10°), recorded from five different populations (indicated by colors). The populations are centered around randomly selected grid locations (see Section Model, subsection Model details). Missing data points imply that there is no good fit of the data for certain stimulus configurations. The star symbols are the mean values of w₁ and w₂ for each population. B, Responses to plaid center stimulus only. C, Responses to plaid center stimulus in the presence of a surround stimulus with orientation equal to the plaid’s second component. Dashed lines are unit diagonals, along which w₁=w₂.

Feature-specific surround suppression A, B, Mean w₁ is plotted against mean w₂ for plaid angles [−60∘,−30∘,30∘,60∘,90∘] from 80 populations, centered around 80 randomly selected grid locations (see Section Model, subsection Model details). The mean values of w₁ and w₂ are obtained from averaging data for different rotations of the plaid (A) for plaid center stimulus only (B) for plaid center stimulus in the presence of surround stimulus with orientation equal the plaid’s second component. C, Mean values of the data points in (A) and (B) for different plaid angles, we also include the data for plaid angle 0°, error bars are the s.e.m.

Activity decay time The time response of an excitatory (E) cell at a randomly selected grid location (see Section Model, subsection Mode details) for various stimulus conditions. A, and B, 2° size stimulus at high contrast (C = 17) and low contrast (C = 9) respectively. C, and D, 10° size stimulus at high contrast (C = 17) and low contrast (C = 9) respectively. The feedforward input is removed at 200 ms. The activity decay time constant is obtained by fitting an exponential function to the decaying activity. The activity decay time constant is roughly independent of the stimulus contrast level and size.

Conductance-based spiking model A, Input currents to an excitatory (E) cell at a randomly selected grid location (see Section Model, subsection Model details) vs external input current. The recurrent excitatory current I_{exc−recurrent} = 〈g_E (R_E − V)〉_t, the recurrent inhibitory current is the absolute value of I_inh = 〈g_I (R_I − V)〉_t and the external current I_ext = 〈g_in (R_E − V)〉_t where 〈〉_t denotes time average. The net current is (I_ext + I_{exc−recurrent} + I_inh), note that I_inh is negative in the spiking model. All currents are normalized to the peak value of the recurrent excitatory current. Stimulus contrast level corresponding to the external current is shown on the bottom axis. B, Firing rate of the cell in (A) vs contrast. C, I_{exc−recurrent}/(I_{exc−recurrent} + |I_inh|) vs contrast for the cell in (A) and an inhibitory cell at the same grid location. In these experiments we use a stimulus with width 2.16° and orientation equal to the cell’s preferred orientation.

Conductance-based spiking model, surround suppression size tuning curves of 6 excitatory (E) cells (top panel) and 6 inhibitory (I) cells (lower panel) for two different stimulus contrast levels C = 100 and C = 10. The error bars are the s.e.m.

Conductance-based spiking model, surround suppression The average firing rate of 30 excitatory (E) cells at randomly selected grid locations (see Section Model, subsection Model details), and of 30 inhibitory (I) cells at the same grid locations, after normalizing each cell’s rates so that its peak rate is 1.0, vs. stimulus size at contrast C = 100 (A) and contrast C = 10 (B). C, D, Summation Field Sizes. C, The distribution of summation field size of the 30 E cells used to produce panels (A) and (B) at contrasts C = 100 (dark red color) and C = 10 (light red color). D, The distribution of the ratio of the summation field sizes in (C). The summation field size of all cells, is smaller at the higher contrast stimulus.

Conductance-based spiking model, surround suppression Excitatory and inhibitory conductances values of the 30 excitatory (E) cells used in Figure 14, g_e and g_i are the time average values of the excitatory and inhibitory conductances in Equation 7, respectively. A, Excitatory conductance values of the E cells for a large suppressive stimulus are plotted against their values for a small stimulus size around which the cells respond maximally (we pick the small stimulus size using the method described in Section Results, subsection Surround suppression). B, same as (A) but for inhibitory conductances. Stimulus contrast C = 100.