Das & Gilbert: G01364A HP/mwSupplementary InformationPg. 1

Supplementary Information, accompanying:

Topography of contextual modulations resulting from short-range interactions in primary visual cortex

Aniruddha Das and Charles D. Gilbert

Model of flank suppression mediated through common input

We wish to address here how inhibitory interactions between two neurons, as seen in the suppressive effect of flanking stimuli, is consistent with the excitatory effective physiological interconnections between them as manifested by significant positive and narrow cross correlations (Fig. 3, main paper). Here we present a physiologically plausible model of local circuitry that could account for these apparently contradictory observations. Consider an experimental configuration as in Fig. 3 of the main paper, recording simultaneously from a test neuron and a flank neuron in neighbouring, orthogonal cortical columns. The model circuit is shown in Fig S1a. T and F represent the test and flank neurons respectively, separated by an orientation singularity (dashed line). Two distinct populations of neurons, CF and CT, provide local common input to F and T: CF represents other neurons in the same column as F, while CT represents other neurons in the same column as T. CF is shown connected to both F and T, with both a direct excitatory input and an inhibitory input mediated through an inhibitory interneuron. CT is assumed to have identical, mirror-image circuitry, (not shown in order to avoid clutter in the figure). Both F and T could also have purely excitatory local circuits of self-amplification1,2 (shown explicitly only for F in the figure). The modulation in instantaneous firing probability effected by common input is represented by the profile shown in Fig. S1b – an excitatory increase of amplitude E and duration E, followed by an inhibitory decrease of amplitude I and duration I3. The profile of inhibition could be flat over time, or be stronger initially (indicated by I’; see Fig. S2d) This interaction profile is assumed to be identical for all the common input connections, i.e. from both CT and CF to both F and T. The effect of self-amplification is represented as an amplification of the excitatory amplitude of the common input profile, from E to AE as shown in Fig. S1b (A: a multiplicative factor greater than or equal to 1.0; possible model values for this will be discussed later.). This amplification is assumed to occur only for the “same column” common inputs, i.e. for CF to F and CT to T. The nominal stimulus-driven firing rates in the different units are given by RT and RF for the test and flank units respectively, and RCT and RCF for the “test-column common input” and “flank-column common input” respectively. Since CT only “sees” the same stimulus as T, its spike rate RCT is assumed to be proportional to RT, and set to frT . f is a fraction less than or equal to 1.0, on the assumption that while many thousands of neurons may be engaged in providing common input, only the small subset of coincident spikes would be effective. This quantity f will be elaborated upon later. From similar considerations, RCF is set to fRF.

Spike trains were simulated in the four “neuron” units using four random number generators with homogenous distributions from 0->1. Each unit was assigned firing probabilities as indicated in Fig S1c. The time axis was divided into equal bins of length  (=0.25 ms). To start a sweep, each unit was assigned a constant initial firing probability based on the nominal instantaneous firing rate. Thus, e.g., the probability of getting a spike in unit T in the first time bin was set = RT (=0.00025RT). Random numbers were generated, one for each unit, and a spike was registered in that time bin for a given unit if the number generated for that unit was less than the corresponding probability. Over the course of a simulated sweep the firing probabilities for the two common input units were kept constant (= RCT and RCF respectively). Each time a spike was registered in either of the common input units, however, the probability distributions in the test and flank units were modified by adding the interaction profile starting at the next time bin, as shown in Fig. S1c (arrow), to simulate the modulatory effect of the common input. Two stimulus configurations were simulated: “Stim Alone”, with RT high and RF low, and “Stim+Flank” with the same RT but with RF high. The model was run using various combinations of model parameters (E, I, I’, E, I, f, A). 20 sweeps of 2 seconds each were simulated in each run and the simulated spike trains from units T and F analysed using exactly the same procedures as were used for the real experimental data. PSTHs were obtained for each 2-second sweep, the total number of test spikes for the “Stim Only” and “Stim+Flank” conditions compared, and the spike trains for T and F cross correlated.

The spike trains simulated using the above model showed clear cross correlation despite strong suppression (of 15% to 74% depending on the choices of model parameters) with high “flank” spike rate. Four representative examples with different model parameters (and flank-induced suppressions ranging from 26% to 74%) are illustrated in Fig. S2. The particular model parameters used, and the consequent degree of simulated flank suppression as well as correlation strength lead to suggested experimental tests of the underlying physiological parameters. These are described in some detail below.

  • In an initial set of simulations we assumed no self amplification (A=1.0), with each common unit providing identical common input to F and T. Despite a large range of excitatory time constants E (5 ms. to 20 ms.) and inhibitory time constants I (50 ms. to 200 ms.3) used, simulated flank suppression could not be pushed beyond about 15%. The reason for this is that any choice of parameters adequate to give strong suppression during “Stim+Flank” also gave strong self-suppression of the test signal through the “test-column” common units, leading to low “Stim Alone” rates as well.
  • With self amplification factors ranging from A=2.0 to A=4.0, we simulated flank-induced suppressions of 26% to 35% (Fig. S2a; with I = 187 ms.). These larger values could be obtained because now there was an asymmetry between a common unit’s input to its own column and to the orthogonal column. The inputs from CT and CF were mirror-symmetric, however. This resulted in both F and T being suppressed to a similar degree and having similar simulated final rates for the “Stim+Flank” configuration. (Note the similarity between the test and flank histogram values for the S+F in each case). For similar reasons, stronger values of flank-induced suppression could not be achieved since increasing values of the inhibition parameters resulted in stronger self-suppression of the flank as well.
  • We next simulated an inhibitory amplitude that was not fixed but, rather, varied with the average common input spike rate. (Experimentally, inhibition due to interneurons in local upper-layer V1 circuits is known to need a higher threshold to get activated, after which it increases in amplitude much faster than excitation; electrical stimulation of local interactions in V1 show only excitation when the stimulus strength is low. With stronger stimuli, inhibition can be seen, while with even stronger stimuli inhibition dominates the post-synaptic modulation3). This was simulated by having the inhibition amplitude I =I0(R/R0)2, non-linearly dependent on the average spike rate R in the relevant channel while the excitation amplitude E was maintained constant. R0 was chosen = 30 spikes/sec for both the test and flank channels. Using this model of inhibition, and with an initial flank spike rate RF that was 2X the test spike rate RT we could simulate suppressions of 65% to 75% (Fig. S2b). The need for a flank stimulus that is stronger than the test stimulus is consistent with our observation that the flank stimulus needed to be of a higher contrast than the test stimulus for significant suppression of the signal.
  • The problems with the above simulations can be seen in Fig. S2b. Despite the asymmetry between the rates RF and RT and the consequently stronger inhibition in the common inputs from F to T, the suppression of F is also strong (see the flank histogram for S+F). This is because the strong inhibition of F on T, mediated through CF, also acts on and suppresses F. The final simulated flank rate ~ 40 spikes/sec. is inconsistent with the high rates required in F to activate the non-linearly higher inhibition in this model. Increasing the self-amplification term here leads to worsening correlation between the simulated spike trains in T and F. One possible resolution of this is explored in Fig. S2c. Here, we considered the possibility that the fraction f of effective spikes that feed the common input circuit was not a constant fraction of the spike rate in the relevant channel but, rather, non-linearly dependent on the spike rate. This is consistent with the idea that spikes are more effective if they are coincident. This implies a dependence on the square of the relevant spike rate: the effective fraction f = f0(R/R0)2. On using this simulated non-linearity in the fraction of spikes from either channel that enter the common input circuit (R0= 30 spikes/sec), we obtained results such as in Fig. S2c, where a nominal RF of 100 spikes/sec. gave a final simulated spike rate of 90 spikes/sec. in the flank channel, with 68% reduction in the test channel.
  • Finally, an introduction of a stepped profile for the IPSP, with an initial dip 2X to 4X as large as the sustained inhibition and lasting 25 ms, led to the appearance of inhibitory side bands around the cross-correlation peak, as well as a degree of asymmetry in the cross correlation peak (Fig.S2d), akin to the features seen in the experimental results. This increased dip could be a reflection of the time course of the simulated IPSP. It could also be a simulation of the influence of an orientation-independent source of inhibition, such as local basket cells, providing an additional burst of inhibition that is tied to excitatory inputs but delayed by one synapse in time.

We have used in this model several features of cortical circuits that incorporate experimental results or that suggest plausible and experimentally verifiable features of synaptic effectiveness. These include strong common input interactions between nearby columns, which is consistent with the anatomy; a changing balance between inhibition and excitation as spike rate is increased; a measure of self amplification in local cortical circuits; and an increase in synaptic effectiveness as spike rate is increased. The result is a circuit that allows for resultant orientation-specific inhibitory flanks while showing correlations of the “excitatory” common input type, thus reproducing our observations (Fig. 3 of main paper).

References

1: Douglas, R.J., Koch, C., Mahowald, M., Martin, K.A.C. and Suarez, H. Recurrent excitation in neocortical circuits. Science269, 981-895 (1995)

2: Somers, D.C., Nelson, S.B. and Sur, M. An emergent model of orientation selectivity in cat visual cortical simple cells. J. Neurosci. 15, 5448-5465. (1995)

3: Hirsch, J. and Gilbert, C.D. Synaptic physiology of horizontal connections in the cat’s visual cortex. J. Neurosci.11(6), 1800-1809 (1991)

Figure S1: Model of local cortical circuit providing common excitatory and (delayed) inhibitory input, for simulating both the flank-induced suppression of neuronal responses, and the strong observed cross-correlation. a: Model circuit . See text of appendix for details. b: Profile of modulatory input provided by the model common input units. c: Model of spike firing probability shown over successive time bins for the test and flank units. For each probability profile, a spike in one of the common input units causes a modulation of the profile starting at the time bin marked (arrow).

Figure S2: Simulation results for four representative ranges of model parameters. a: Self-amplification of 4.0; b: In addition to self-amplification, an inhibition strength that increases with increasing neuronal spike rate; c: In addition to the parameters in (b), the effective fraction of spikes forming the common input increases with average firing rates; d: In addition to the parameters in (c), the inhibition profile is stepped with an initially higher inhibition I’ lasting 25 ms. See text for details about the different model parameters for the four simulations shown. Each set of results gives the response histograms (left) and shift-corrected, normalized cross-correlogram (right). Response histograms were averaged from PSTHs of simulated spike trains, 20 sweeps for each stimulus condition (S: Stim alone; S+F: Stim+Flank), each sweep of 2 sec. duration. Upper (black): test, lower (grey): flank. For 7a RT was set = 100 spikes/sec for both stimulus conditions while for 7b through 7d RT was set = 50 spikes/sec. “Stim Alone” was simulated by setting RF = 5spikes/sec. in all examples, while “Sim+Flank” was simulated by setting RF = 100 spikes/sec. Cross-correlograms were obtained from the “S+F” spike trains alone; the thin grey line shows the average shift predictor in each case. Each panel lists the height and area of the normalized cross-correlogram peak, as well as the strength of the test signal suppression by the flank (calculated as in the real experiments, signal at “S+F” – signal at “S”, expressed as a fraction of the signal at “S”; the “spontaneous” was taken = 5 spikes / sec.).