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On the potential role of lateral connectivity in retinal anticipation - PubMed

  • ️Fri Jan 01 2021

On the potential role of lateral connectivity in retinal anticipation

Selma Souihel et al. J Math Neurosci. 2021.

Abstract

We analyse the potential effects of lateral connectivity (amacrine cells and gap junctions) on motion anticipation in the retina. Our main result is that lateral connectivity can-under conditions analysed in the paper-trigger a wave of activity enhancing the anticipation mechanism provided by local gain control (Berry et al. in Nature 398(6725):334-338, 1999; Chen et al. in J. Neurosci. 33(1):120-132, 2013). We illustrate these predictions by two examples studied in the experimental literature: differential motion sensitive cells (Baccus and Meister in Neuron 36(5):909-919, 2002) and direction sensitive cells where direction sensitivity is inherited from asymmetry in gap junctions connectivity (Trenholm et al. in Nat. Neurosci. 16:154-156, 2013). We finally present reconstructions of retinal responses to 2D visual inputs to assess the ability of our model to anticipate motion in the case of three different 2D stimuli.

Keywords: 2D; Lateral connectivity; Motion anticipation; Retina.

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Conflict of interest statement

The authors declare that they have no competing interests.

Figures

Figure 1
Figure 1

Synthetic view of the retina model. A stimulus is perceived by the retina, triggering different pathways. Pathway I (blue) corresponds to a feed-forward response where, from top to bottom: The stimulus is first convolved with a spatio-temporal receptive field that mimics the outer plexiform layer (OPL) (“Bipolar receptive field response”). This response is rectified by low voltage threshold (blue squares). Bipolar cell responses are then pooled (blue circles with blue arrows) and input ganglion cells. The firing rate response of a ganglion cell is a sigmoidal function of the voltage (blue square). Gain control can be applied at the bipolar and ganglion cell level (pink circles) triggering anticipation. This corresponds to the label II (pink) in the figure. Lateral connectivity is featured by pathway III (brown) through ACells and pathway IV (green) through gap-junctions at the level of GCells

Figure 2
Figure 2

Example of a retina grid tiling and indexing. The green and blue ellipses denote respectively the positive centre and the negative surround of the BCell receptive field KS. The centre of RF coincides with the position of the cell (blue and green arrows). The red ellipse denotes the ganglion cell pooling over bipolar cells (Eq. (17))

Figure 3
Figure 3

Gain control (9) as a function of activity A. The function l(A), in dashed line, is a piecewise linear approximation of GB(A) from which three regions are roughly defined. In the region “Silent” the gain vanishes so the cell does not respond to stimuli; in the region “Max”, the gain is maximal so that cell behaviour does not show any difference with a not gain-controlled cell; the region “Fast decay” is the one which contributes to anticipation by shifting the peak in the cell’s activity (see Sect. 3.1). The value Ac=23 corresponds to the value of activity where gain control, in the piecewise linear approximation, becomes effective

Figure 4
Figure 4

The mechanism of motion anticipation and the role of gain control. The figure illustrates the bipolar anticipation time ΔB without lateral connectivity. We see the response of OFF BCells with gain control to a dark moving bar. The curves correspond to three cells spaced by 450μm. The first line (a) shows the linear filtering of the stimulus corresponding to Vdrive(t) (Eq. (1)). Line (b) corresponds to the threshold nonlinearity NB applied to the linear response; (c) represents the adaptation variable (16), and (d) shows the gain control time curse. Finally, the last line (e) corresponds to the response RBi of the BCell. The two dashed lines correspond respectively to tB and tBA, the peak in the response of the (purple) BCell without pooling

Figure 5
Figure 5

Maximum firing rate and anticipation time variability with stimulus parameters in the gain control layer of the model. Left: contrast (with v=1 mm/s et size = 90 μm); middle: size (with v=2 mm/s et contrast = 1); right: speed (with contrast 1 and size = 162 μm)

Figure 6
Figure 6

Front (43) for different values of λβ (purple) as a function of time for the cell γ=0. All figures are drawn with v=2 mm/s; σ=0.2 mm. Top. λβ=−0.5ms−1; Bottom. λβ=0.5ms−1

Figure 7
Figure 7

Anticipation in the Laplacian (nearest-neighbours) case. Top. Anticipation time and maximum bipolar response as a function of the connectivity weight w. The blue line corresponds to gain control alone (it does not depend on w). Red triangles correspond to the effect of lateral ACell connectivity without gain control. White squares correspond to the compound effect of ACell connectivity and gain control. The three regimes A, B, C are commented in the text. Bottom. Response curves of ACells and BCells corresponding to the three regimes: (A) w=0.05ms−1 with a small cross-inhibition, (B) w=0.3ms−1 with an opposition in activity between the blue (cell 50) and red cell (51), (C) w=0.6ms−1, where the red cell (51) is completely inhibited by cell 50

Figure 8
Figure 8

Spectral density of eigenvalues for ξ=2, n¯=4, σn=1. Top left. For the matrix W (density estimated over 10,000 samples), density is represented in colour plots, in log scale. The colour bar refers to powers of 10. Top right. Spectral density of L for w=0.05. Bottom left. Spectral density of L for w=0.1. Bottom right. Spectral density of L for w=0.15. Unstable eigenvalues are on the right to the vertical dashed line x=0

Figure 9
Figure 9

Heat map for the largest real part eigenvalue in the plane w, ξ for different values of n̄. Left. n¯=1. Right. n¯=4. Colour lines are level lines. The level line 0 is the frontier of instability of the linear dynamical system

Figure 10
Figure 10

Average anticipation in the random connectivity case. Top. Bipolar anticipation time and maximum in the response RB as a function of the connectivity weight w in the case of a random connectivity graph with ξ=1, n¯=2 and σn=1. Bottom. Response curves of ACells and BCells 51−52 corresponding to the three regimes: (A) w=0.05ms−1, (B) w=0.3ms−1, (C) w=0.6ms−1

Figure 11
Figure 11

Role of the characteristic branch length ξ on anticipation. Top. The joint effect of the random ACell connectivity and gain control on anticipation for ξ=1,2,3. Left. Average bipolar anticipation time. Right. Maximum value of the bipolar response RB. Bottom. The single effect of the random ACell connectivity on anticipation with the same representation

Figure 12
Figure 12

Anticipation for non symmetric gap junctions. Top. GCell anticipation time and maximum firing rate as a function of the gap junction velocity vgap. Bottom: response curves of GCells corresponding to the three regimes: (A) vgap=0.6mm/s, (B) vgap=3 mm/s, (C) vgap=12mm/s. The curves display 3 main regimes (see text): In (A) vgap<v and the positive front propagates at the same speed as the pooling voltage triggered by the stimulus; In (B), vgap=v, the positive front and the negative fronts both propagate at the speed vgap and the amplitude of the positive front (VG(t)) increases with t; In (C), vgap>v and the positive front propagates faster than the stimulus so that the peak of activity arises earlier. The negative front propagates at the stimulus speed

Figure 13
Figure 13

Anticipation for symmetric gap junctions. Top. GCell anticipation time and maximum firing rate as a function of the gap junction velocity vgap. Bottom. Response curves of GCells corresponding for three values of vgap: (A) vgap=0.6mm/s, (B) vgap=3mm/s, (C) vgap=12mm/s. For consistency, we have kept the same values as in the asymmetric case. Here, anticipation time grows continuously until saturation, while the maximum firing decreases like a power law as a function of vgap

Figure 14
Figure 14

Flash lag effect with different anticipatory mechanisms. (A) Response to a flash lag stimulus: a bar moving in smooth motion with a second bar flashed in alignment with the first bar for one time frame. The first line shows the stimulus, the second line shows the GCells response with gain control, the third line presents the effect of lateral ACell Laplacian connectivity with w=0.3ms−1, and the last line shows the effect of asymmetric gap junctions with vgap=9mm/s. (B) Time course response of (top) a cell responding to the flashed bar and (bottom) a cell responding to the moving bar. Dashed lines indicate the peak of each curve

Figure 15
Figure 15

Effect of anticipatory mechanisms on a parabolic trajectory. (A) Response to a dot moving along a parabolic trajectory. The first line shows the stimulus, the second line shows the GCell response with gain control, the third line presents the effect of lateral ACell connectivity with w=0.3ms−1, and the last line shows the effect of asymmetric gap junctions with vgap=9mm/s. (B) Time course response of a cell responding to the dot near the trajectory turning point. Linear response corresponds to the response to the stimulus without gain control

Figure 16
Figure 16

Anticipation for a rotating bar. (A) Response to a bar rotating at 4.25rad/ms. The first line shows the stimulus, the second line shows the GCell response with gain control, the third line presents the effect of lateral ACell connectivity with w=0.2ms−1, and the last line shows the effect of asymmetric gap junctions with vgap=9mm/s. (B) Time course response of the bar orientation in the reconstructed retinal representations

Figure 17
Figure 17

Filter transformation description [39]. The original system of axes is represented by x and y, and the ellipse system of axes by u and v. ϕ represents the angle of the second axis of the non-orthogonal basis. The integration domain of a pixel is limited by four lines of equations: x=iδ, x=(i+1)δ, y=jδ and y=(j+1)δ. Rewriting these fours equations in the new system of axes through a coordinate change enables us to write Eq. (61)

Figure 18
Figure 18

Random connectivity. Left. Example of a random connectivity matrix from ACells Aj to BCells cells Bi. White points correspond to connection from Aj to Bi. Right. Probability P(d) that two branches intersect as a function of the distance between two cells. ‘Exp’ corresponds to numerical estimation and ‘Th’ corresponds to the theoretical prediction. Here, ξ=2

Figure 19
Figure 19

Geometry of connection between two neurons. α (β) is the angle of the neuron A’s branch with length LA (neuron B’s branch with length LB) with respect to the horizontal axis. θ is the angle between the segment connecting AB and the branch A. C represents the virtual point that lies at the intersection of the branches of length LA and LB. dAB (resp. dAC, dBC) denotes the distance between A and B (resp. A, C and B, C). Note that dAC≤LA, dBC≤LB

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