Summary of key results on different effects of triadic and axonal inhibition on relay-cell (RC) response.

<p>(A) Dependence of diameter of RC receptive-field center on two key model parameters (<i>w</i>_GIp, weight of proximal excitation of the interneuron (IN); <i>w</i>_IRa, weight of axonal inhibition) for the case of axonal inhibition only (i.e., <i>w</i>_IRt = 0). For this example the diameter of the ganglion-cell receptive-field center is fixed to 1.8 deg, and the retinogeniculate excitation is set to <i>w</i>_GR = 11.6 nS. (B) Transient and sustained RC responses for center-filling spots, corresponding to maximal responses in the area-response curves, for the cases of only triadic or only axonal inhibition. Dependence of maximal response on retinogeniculate excitation weight <i>w</i>_GR is depicted. Other parameters: <i>w</i>_IRa = 4 nS, <i>w</i>_GIp = 0.6 nS. (C) Dependence of center-surround antagonism, quantified by the coefficient <i>α</i>_RC (<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004929#pcbi.1004929.e010" target="_blank">Eq 7</a>), on <i>w</i>_GIp, the weight of the GC excitation of INs on the proximal dendrites. Dark-blue line: No inhibition, <i>w</i>_IRa = <i>w</i>_IRt = 0. Red line: Axonal inhibition only, <i>w</i>_IRt = 0, <i>w</i>_IRa = 8 nS. Green line: Triadic inhibition only, <i>w</i>_IRa = 0. Light-blue line: Both triadic and axonal inhibition, <i>w</i>_IRa = 8 nS. Retinogeniculate excitation is set to <i>w</i>_GR = 15.6 nS. In B and C simulation data points are marked with dots, and lines are added as a guide for the eye. Note that 500 trials, not the default value of 10 trials, were used to compute the depicted trial-averaged spike-count rate in panel B.</p

Illustration of area-response curves and metrics used to quantify key properties.

<p>Center diameter <i>d</i>_c, surround diameter <i>d</i>_s, peak response rate <i>r</i>_c, center-surround minimum rate <i>r</i>_<i>cs</i>. Illustration adapted from Fig 1 in [<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004929#pcbi.1004929.ref026" target="_blank">26</a>].</p

Illustration of temporal response in dLGN model circuit.

<p>A stimulus spot of diameter <i>d</i> = 1 deg is turned on at 500 ms. (A) Example of (single-trial) IN membrane-potential dynamics (soma: blue line; distal part of dendritic segment receiving synaptic input from central GC cell: green line). Also shown are GC input spikes driving the circuit, both from the center GC cell (top row of tiny triangles) and from the four peripheral GC cells (bottom row of triangles). (B) Corresponding RC membrane-potential dynamics. Also shown are input spikes from the central GC input (top row of tiny black triangles), IN dendritic (triadic) action potentials (middle row of green triangles), and IN somatic action potentials (bottom row of blue triangles). See text for explanation of arrows. Default model parameters are used, cf. Tables <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004929#pcbi.1004929.t002" target="_blank">2</a>–<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004929#pcbi.1004929.t004" target="_blank">4</a>.</p

Area-response curves without triadic inhibition.

<p>Row 1: no inhibition. Rows 2–4: axonal inhibition for different synaptic weight values of proximal ganglion-cell input to the interneuron <i>w</i>_GIp. Black curves correspond to central retinal ganglion cell (GC), blue curves to interneuron (IN), and red/orange curves to relay cell (RC). The four RC curves in the panels in rows 2–4 correspond to <i>w</i>_IRa = 0/2/4/8 nS with <i>w</i>_IRa = 0 (no axonal inhibition) and <i>w</i>_IRa = 8 nS (maximal axonal inhibition) corresponding to the top and bottom of the four curves, respectively.</p

Schematic of the dLGN circuit model.

<p>(Top) Five relay cells (RCs) receive input from one retinal ganglion (GC) cell each. All inputs to RCs arrive in triadic synapses, involving the one and same IN. In addition, the IN receives proximal input from all five GCs. The boxes highlight the synaptic connections in the networks and the associated connection weights <i>w</i>. Note that in the present model application, only responses for the central RC cell is considered so that the only effect of the four peripheral GCs comes from the proximal inputs to the IN. (Bottom) The GCs are organized with four peripheral GCs all located at distance <i>r</i>_a from the center GC.</p

Summary of response measures from area-response curves with triadic inhibition present.

<p>(A) Ratio of receptive-field center diameter of relay cell (RC) and (central) retinal ganglion cell (GC), ; receptive-field center diameter measured as the spot diameter corresponding to the largest firing rate in the area-summation curves in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004929#pcbi.1004929.g009" target="_blank">Fig 9</a>. (B) Ratio of receptive-field center diameter of interneuron (IN) and relay cell (RC), . (C) Relay-cell <i>α</i>_R (solid) and interneuron <i>α</i>_I (dashed) center-surround antagonisms, cf. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004929#pcbi.1004929.e010" target="_blank">Eq 7</a>. (D) Maximal firing rate <i>r</i>_c, i.e., firing rate for spot exactly covering receptive-field center, for relay cell (, solid) and interneuron (, dashed). The colored lines correspond to different values of <i>w</i>_GIp, see legend below panels. Note also that interneuron (IN) results are absent for the case with <i>w</i>_GIp = 0 (blue lines) since in this case the IN only receives triadic input and does not fire any action potentials.</p

Area-summation curves for transient response and sustained response for relay-cells (RCs).

<p>(A–B) Trial-averaged spike-count firing rate vs. spot diameter, for central retinal ganglion cell (GC, solid black), interneuron (IN, solid blue), and relay cell (RC, red lines) for transient (A) and sustained responses (B). (C–D) Area-summation curves in A and B normalized to the maximum firing rate for each cell. The transient response corresponds to the trial-averaged spike-count firing rate for the first 100 ms after stimulus onset, while the sustained response corresponds to the averaged rate in the time interval from 400 to 500 ms after stimulus onset, cf. <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004929#pcbi.1004929.g007" target="_blank">Fig 7</a>. The depicted models examples are the same as in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004929#pcbi.1004929.g008" target="_blank">Fig 8</a>: Red solid line: RC response for <i>direct triadic inhibition</i> (case (RC-i)) with <i>w</i>_IRt = 4 nS, <i>w</i>_GIp = 0, <i>w</i>_IRa = 0. Red dashed line: RC response for <i>direct & soma-driven triadic inhibition</i> (case (RC-ii)) with <i>w</i>_IRt = 4 nS, <i>w</i>_GIp = 0.6 nS, <i>w</i>_IRa = 0. Red dotted line: RC response for <i>axonal inhibition</i> (case (RC-iii)) with <i>w</i>_IRt = 0, <i>w</i>_GIp = 0.6 nS, <i>w</i>_IRa = 4 nS. Dark red line (RC-all) corresponds to results from all three types of inhibition combined, i.e., <i>w</i>_IRt = 4 nS, <i>w</i>_GIp = 0.6 nS, <i>w</i>_IRa = 4 nS. <i>w</i>_GR = 15.6 nS is used in all cases. Other parameters correspond to default values. Note that the depicted IN response does not apply to case (RC-i) as the IN is only synaptically activated at the triads in this case as <i>w</i>_GIp = 0. Note also that 500 trials, not the default value of 10 trials, were used to compute each depicted trial-averaged spike-count rate, and that no seven-point filtering was employed to smooth the area-summation curves.</p

Synaptic integration properties of interneuron (IN) model.

<p>(A) Ball-and-sticks IN model consisting of a point-like soma (black square) with five dendritic sticks protruding out from it. Distal (triadic; blue dots) and proximal (red dots) synapse locations are illustrated. Panels B–E shows spatiotemporal spread of IN membrane potential along two (of five) dendritic sticks following activation by single RC spiking on inputs distal (triadic) and/or proximal synapses. Each colored line represents a snapshot of the membrane potential taken each half millisecond from 0 to 20 milliseconds with the GC spike(s) arriving at <i>t</i>_syn = 1 ms. The synapse position(s) are denoted by vertical, red or blue dashed lines, while the black dashed line marks the location of the soma compartment. The small inset axes show the membrane potential in the soma (<i>V</i>_<i>i</i>(0, <i>t</i>)) and in the distal dendrite (<i>V</i>_<i>i</i>(−450 <i>μ</i>m, <i>t</i>)), respectively, as a function of time. (B) GC spike onto distal synapse on lower left dendritic stick. (C) GC spike onto proximal synapse on lower left dendritic stick. (D) GC spike arriving simultaneously on distal and proximal synapses on lower left dendritic stick. (E) GC spikes arriving simultaneously at all five proximal synapses, including those on the two depicted dendritic sticks.</p

Example post-stimulus time histograms (PSTS) for cells in dLGN model circuit.

<p>Stimulus spot of diameter <i>d</i> = 1 deg is turned on at 500 ms. (A) PSTH for central GC cell. (B) PSTH for IN cell. (C) PSTH for RC cell with axonal inhibition only (<i>w</i>_IRa = 4 nS, <i>w</i>_IRt = 0). (D) PSTH for RC cell with triadic inhibition only (<i>w</i>_IRa = 0, <i>w</i>_IRt = 4 nS). Results correspond to 1000 trials, bin size: 5 ms. Default model parameters are used, cf. Tables <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004929#pcbi.1004929.t002" target="_blank">2</a>–<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1004929#pcbi.1004929.t004" target="_blank">4</a>.</p

Parameter space explored for spot size and synaptic weights (maximal synaptic conductances, cf. Eq 6) in simulations.

<p>† denotes default values.</p