Virtual Retina: A biological retina model and simulator, with contrast gain control

Abstract

We propose a new retina simulation software, called Virtual Retina, which transforms a video into spike trains. Our goal is twofold: Allow large scale simulations (up to 100,000 neurons) in reasonable processing times and keep a strong biological plausibility, taking into account implementation constraints. The underlying model includes a linear model of filtering in the Outer Plexiform Layer, a shunting feedback at the level of bipolar cells accounting for rapid contrast gain control, and a spike generation process modeling ganglion cells. We prove the pertinence of our software by reproducing several experimental measurements from single ganglion cells such as cat X and Y cells. This software will be an evolutionary tool for neuroscientists that need realistic large-scale input spike trains in subsequent treatments, and for educational purposes.

Appendices

Appendix A - Coupling kernel in a layer of cells

Let us comment further the choice of the convolution kernel G _σ E _τ (x,y,t) to model signal averaging in a layer of cells (see Section 2.1.3). Suppose that a layer of retinal cells, described by the spatially continuous potential V(x,y,t), is linearly driven by an input synaptic current I(x,y,t). Then, V can always be linearly calculated from I, through an impulse response K(x,y,t):

$$V(x,y,t)= K * I(x,y,t),$$

(19)

where symbol * represents spatio-temporal convolution. Because neurons are small ‘RC’ circuits, K is temporally low-pass, with a term in $\exp(-t/\tau)$. However, the precise expression of K depends on the type of spatial averaging being modeled. There are two effects:

• Averaging because of the cells’ dendritic spread is well modeled by a static spatial Gaussian kernel, leading to the separable filter

  $$K_{\textrm{dendritic spread}} (x,y,t) = G_\sigma(x,y) \exp(-t/\tau).$$

  (20)

• Averaging by gap junctions between neighboring cells can be expressed either in a discrete-cell approach (Mahowald and Mead 1991; Herault 1996) or in a continuous setting with Laplacian-like operators (Naka and Rushton 1967; Lamb 1976). Both approaches lead approximately to the same impulse response

  $$K_{\textrm{gap junctions}} (x,y,t) = G_{\displaystyle \sqrt{2\mathcal{G}t}}(x,y) \exp(-t/\tau),$$

  (21)

  where $\mathcal{G}$ is a constant measuring the two-dimensional density of gap junctions.

One can verify that both Eqs. (20) and (21) are spatio-temporal low-pass filters, with very similar characteristics in the Fourier domain. Filter Eq. (21) is a bit harder to handle mathematically, because of the influence of time t on the spatial Gaussian kernel. For this reason, in this article we model all low-pass effects, including effect of gap junctions, with separable filters like Eq. (20). However, the filtering kernel Eq. (21) is also implemented in Virtual Retina.

Appendix B - Mathematical analysis of the contrast gain control loop

An original contribution in this work was the proposition of a contrast gain control mechanism (Section 2.3) via the differential equation

$$\frac{\displaystyle {dV}}{\displaystyle {dt}}(x,y,t)=I_\mathrm{OPL}(x,y,t) -g_\mathrm{A}(x,y,t) V(x,y,t)$$

(22)

with

$$g_\mathrm{A}(x,y,t) = G_{\sigma_\mathrm{A}} \stackrel{x,y}{*} E_{\tau_\mathrm{A}} \stackrel{t}{*} Q (V) \;\;(x,y,t),$$

(23)

$$Q(V) = g^0_\mathrm{A} + \lambda_\mathrm{A} V^2.$$

(24)

Mathematically, this dynamical system is difficult to study due to its high dimensionality (two variables V and g _A , expressed on spatial maps). Thus, in Wohrer (2007), we studied the simplified dynamical system

$$\frac{\displaystyle {dV}}{\displaystyle {dt}}(t)= {A} \cos (\omega t) - V(t)Q(V(t)),$$

(25)

for which we can prove contrast gain control properties. System (25) derives from (22) considering the following assumptions:

• We considered a sinusoidal stimulation: I _OPL(t) = A cos(ωt). This is the simplest way to control both amplitude A (i.e., contrast) and speed of temporal variation ω in the input. Furthermore, sinusoidal stimulation enables direct comparison of our system with linear ones, for which Fourier analysis can be done.

• We assumed that σ _A = 0, so that (25) depends only on time t, and not on any spatial structure (x,y). This choice does not appear too restrictive, especially since contrast gain control is experimentally settled as a temporal property only (see Section 4.2).

• We consider the asymptotic limit of Eq. (22), when parameter τ _A in (23) tends to zero, yielding g _A(t) = Q(V _Bip(t)). As a consequence, (25) is now a one-dimensional dynamical system, easier to study. The assumption τ _A ≃ 0 is justified in the scope of our simulations, for which we chose a small constant τ _A = 5 ms, as detailed in Sections 3.2.2 and 4.2.

In Wohrer (2007), we proved general properties of system (25).

First, (25) is a very stable system. Similarly to a simple linear exponential filter, the system (25) forgets its initial condition exponentially fast (Lohmiller and Slotine 1998): All trajectories converge asymptotically to a unique solution V(t) which is $\frac{2\pi}{\omega}$-periodic (like the input current).

Furthermore, over one cycle, V(t) reaches a single maximum V _max at time t _max, and a single minimum V _min = − V _max at time t _min = t _max − π/ω. In Wohrer (2007), we studied V _max as a measure of the strength of the system’s response to the input current, and ωt _max as a measure of the phase of the system’s response to the input current.

By studying V _max and t _max, we thus provided a description of the system’s behavior according to input frequency and amplitude. This is presented in the following theorem, which shows that (25) acts as a low-pass, gain control system on its input, given suitable assumptions on Q.

Theorem 1

Let V be a solution of (25), with Q an even, convex and strictly positive function. First, we show how V_max and ωt_max depend on the frequency ω:

• (i) Low-pass setting

  $$\partial_{\omega}{V_\mathrm{max}}< 0\,\;and\,\;\displaystyle \lim_{\omega \to +\infty} V_\mathrm{max} = 0.$$

• (ii) Phase delay

  $$\partial_{\omega}{(\omega{t_\mathrm{max}})}> 0\,\;and\,\;\displaystyle \lim_{\omega \to +\infty} \omega{t_\mathrm{max}} = \frac{\pi}{2}\; \textrm{{\rm(}mod $2\pi${\rm)}}.$$

  Second, we show how V _max and ωt _max depend on the amplitude A:

• (iii) Growth of V _max

  $$\partial_{{A}}{V_\mathrm{max}} > 0\,\;and\,\;\displaystyle \lim_{{A} \to +\infty} V_\mathrm{max} = +\infty.$$

• (iv) Phase advance

  $$\partial_{{A}}{t_\mathrm{max}} < 0\,\;and\,\;\displaystyle \lim_{{A} \to +\infty} \omega{t_\mathrm{max}} = 0 \; \textrm{(mod $2\pi$)}.$$

• (v) Under-linearity

  $$\begin{array}{rl} \forall\; (\omega, Q),\;&if\;A\;is\;high\;enough,\;\partial_{{A}}{\frac{V_\mathrm{max}}{{A}}} < 0.\\ &Also,\;\displaystyle \lim_{{A} \to +\infty} \frac{V_\mathrm{max}}{{A}}=0. \end{array}$$

Let us comment those results.

Properties (i) and (ii) show that system (25) acts as a low-pass temporal filter (using a classical linear systems terminology). To understand this, suppose that $Q(V)=g_\mathrm{A}^0$, i.e., just a constant function. This means having a null feedback strength λ _A in Eq. (24). Then, Eq. (25) becomes a simple exponential low-pass filter

$$\dot{V}(t)={A} \cos(\omega t) - g_\mathrm{A}^0 V(t),$$

(26)

whose behavior is well known:

$$V_\mathrm{max} =|\widetilde{H}(\omega)|{A}$$

(27)

$$\omega t_\mathrm{max} =\arg(\widetilde{H}(\omega)),$$

(28)

where $\widetilde{H}(\omega)=1/(g_\mathrm{A}^0+\textbf{j}\omega)$ is the Fourier transform of the system. One verifies easily that Eqs. (27)–(28) imply that V _max tends to 0 (with V _max > 0) and ωt _max tends to $\frac{\pi}{2}$ ($\omega{t_\mathrm{max}}<\frac{\pi}{2}$), as ω→ + ∞. Properties (i)–(ii) extend these properties in the nonlinear case of (25).

Properties (iii), (iv) and (v) show the apparition of gain control in system (25), as opposed to the linear case (26). In the linear case, Eqs. (27)–(28) give a linear dependence of V _max and t _max with respect to amplitude A. One has indeed $\partial_{{A}}{V_\mathrm{max}}=|\widetilde{H}(\omega)|>0$, $\partial_{{A}}{(V_\mathrm{max}/{A})}=0$ and $\partial_{{A}}{(\omega t_\mathrm{max})}=0$. In the nonlinear case (25), property (iii) shows that V _max is still a growing function of A. However, this growth is now under-linear (property (v)).

Finally, property (iv) shows the phase advance effect, with input amplitude A. This second nonlinear effect corresponds to the definition of contrast gain control in real retinas, as detailed in Section 3.2.2.

Appendix C - Example of an XML configuration file

Figure 12 is a snapshot from an XML definition file used in the simulator (with basis parameters for the X cell modeled in Fig. 5(a)). Various XML nodes – such as contrast gain control or the spiking array – can be present or absent from the file, adapting the simulated model’s complexity in consequence.

Fig. 12

Sample XML file used by Virtual Retina to load the architecture and parameters of the used model. This file is highly modular

In particular, a radial scheme can be present or not (here it is useless because a single cell is modeled). The spiking array can either be square, or made of concentric circles following retinal density (as here). In this example file, given the parameters of the circular spiking array, a single spiking cell is present (as used for reproduction of physiological recordings).