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A single retinal circuit model for multiple computations

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Abstract

Vision is dependent on extracting intricate features of the visual information from the outside world, and complex visual computations begin to take place as soon as at the retinal level. In multiple studies on salamander retinas, the responses of a subtype of retinal ganglion cells, i.e., fast/biphasic-OFF ganglion cells, have been shown to be able to realize multiple functions, such as the segregation of a moving object from its background, motion anticipation, and rapid encoding of the spatial features of a new visual scene. For each of these visual functions, modeling approaches using extended linear–nonlinear cascade models suggest specific preceding retinal circuitries merging onto fast/biphasic-OFF ganglion cells. However, whether multiple visual functions can be accommodated together in a certain retinal circuitry and how specific mechanisms for each visual function interact with each other have not been investigated. Here, we propose a physiologically consistent, detailed computational model of the retinal circuit based on the spatiotemporal dynamics and connections of each class of retinal neurons to implement object motion sensitivity, motion anticipation, and rapid coding in the same circuit. Simulations suggest that multiple computations can be accommodated together, thereby implying that the fast/biphasic-OFF ganglion cell has potential to output a train of spikes carrying multiple pieces of information on distinct features of the visual stimuli.

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Authors and Affiliations

  1. Department of Advanced Analytics, Supply Chain Wizard LLC, 34870, Istanbul, Turkey

    Murat Sağlam

  2. Graduate School of Engineering, Osaka University, Suita, Osaka, 565-0871, Japan

    Yuki Hayashida

Authors
  1. Murat Sağlam
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  2. Yuki Hayashida
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Corresponding authors

Correspondence to Murat Sağlam or Yuki Hayashida.

Additional information

Communicated by J. Leo van Hemmen.

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Supplementary material 2 (M4 V 2392 kb)

Appendix

Appendix

1.1 Details of the cellular dynamics

1.1.1 Photoreceptors

The input to the retina model was applied at the photoreceptor level. Each photoreceptor on the two-dimensional spatial grid receives an input of light intensity \( l_{{\underline{x} ,0}} \left( t \right) \) that is processed by a cascade of transformations shown in Eq. (2) as explained previously (Thiel et al. 2006). First, \( l_{{\underline{x} ,0}} \left( t \right) \) was smoothened by a low-pass filter with a time constant \( \tau_{\text{P}} \), and the resulting signal \( l_{{\underline{x} ,1}} \left( t \right) \) was then processed by a compressive nonlinearity yielding a signal \( l_{{\underline{x} ,2}} \left( t \right) \) that has a wider dynamic range than \( l_{{\underline{x} ,1}} \left( t \right) \). The \( l_{{\underline{x} ,2}} \left( t \right) \) was further smoothened by a cascade of two low-pass filters with time constants \( \tau_{\text{P}} \) to obtain the preprocessed photoreceptor input \( l_{{\underline{x} ,{\text{P}}}} \left( t \right) \).

$$\begin{aligned}& l_{{\underline{x} ,1}} \left( t \right) = \left( {1 + \tau_{\text{P}} \frac{\text{d}}{{{\text{d}}t}}} \right)l_{{\underline{x} ,0}} \left( t \right), l_{{\underline{x} ,2}} \left( t \right) = R\left( {\frac{{l_{{\underline{x} ,1}} \left( t \right)}}{{l_{{\underline{x} ,1}} \left( t \right) + I_{0} }}} \right)^{r} , \\& l_{{\underline{x} ,{\text{P}}}} \left( t \right) = \left( {1 + \tau_{\text{P}} \frac{\text{d}}{{{\text{d}}t}}} \right)\left( {1 + \tau_{\text{P}} \frac{\text{d}}{{{\text{d}}t}}} \right)l_{{\underline{x} ,2}} \left( t \right)\end{aligned} $$
(2)

Each photoreceptor was modeled as having two parts, and Eq. (1) governed the dynamics of either part. The membrane potential response of the first part was indicated by \( v_{{\underline{x} ,{\text{P}}}} \left( t \right) \) and stimulated by \( l_{{\underline{x} ,{\text{P}}}} \left( t \right) \) with a delay of \( \Delta t_{\text{P}} \), and thus \( e\left( t \right) \), for this first part, was \( l_{{\underline{x} ,{\text{P}}}} \left( {t - \Delta t_{\text{P}} } \right) \). The second part was dedicated to a synaptic complex with the triad synapse composed of the photoreceptor axon terminal and the dendrite tips of horizontal and bipolar cells. In addition, for this part, the input signals \( e\left( t \right) \) and \( i\left( t \right) \) were provided as \( w_{\text{C}}^{\text{P}} v_{{\underline{x} ,{\text{P}}}}^{ + } \left( t \right) \) and \( w_{\text{C}}^{\text{P}} v_{{\underline{x} ,{\text{P}}}}^{ - } \left( t \right) \), respectively, where \( v_{{\underline{x} ,{\text{P}}}}^{ + } \left( t \right) \) and \( v_{{\underline{x} ,{\text{P}}}}^{ - } \left( t \right) \) stand for the membrane potential response of the first part, and the response to these inputs was indicated by \( v_{{\underline{x} ,{\text{C}}}} \left( t \right) \). Since photoreceptors receive delayed inhibitory feedback signals via the triad synapses (Baylor et al. 1971), \( i\left( t \right) \) for the first part of the photoreceptor was provided as \( w_{\text{P}}^{\text{C}} v_{{\underline{x} ,{\text{C}}}}^{ - } \left( {t - \Delta t_{\text{H}} } \right) \).

1.1.2 Horizontal cells

The horizontal cells receive both depolarizing and hyperpolarizing inputs from photoreceptors via the triad synapse. Hence, \( e\left( t \right) \) and \( i\left( t \right) \) of each horizontal cell were provided as \( w_{\text{H}}^{\text{C}} v_{{\underline{x} ,{\text{C}}}}^{ + } \left( t \right) \) and \( w_{\text{H}}^{\text{C}} v_{{\underline{x} ,{\text{C}}}}^{ - } \left( t \right) \), respectively. The gap junctions among the neighboring cells were realized by the term \( \sum\nolimits_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{j} = \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}^{1} }}^{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{x}^{6} }} {w_{\text{H}}^{\text{H}} } v_{{\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{j} ,{\text{H}}}} \left( t \right) \) in Eq. (1), where \( w_{\text{H}}^{\text{H}} \) determines the signaling weight (i.e., time constant for charging the membrane through the gap junction conductance) and \( v_{{\underline{j} ,{\text{H}}}} \left( t \right) \) is the membrane potential response of a neighboring cell. Each horizontal cell was adjacent to six other horizontal cells on the hexagonal spatial grid of the model (Fig. 1, distal layer); therefore, \( \underline{j} \) was \( \underline{x}^{1} , \underline{x}^{2} , \ldots \underline{x}^{6} \).

1.1.3 Bipolar cells

The ON- and OFF-types of bipolar cells depolarize or hyperpolarize in response to light increases, respectively. The input to the OFF-bipolar cell via the triad synapse was provided as

$$ v_{{\underline{x} ,{\text{C}}}} \left( t \right) - v_{{\underline{x} ,{\text{H}}}} \left( t \right). $$

The sign-conserving term from the photoreceptor and sign-inversing term from the horizontal cell are essential to realizing the well-known, center-surround antagonistic spatial profile of the bipolar cell receptive field. In addition, each bipolar cell was modeled as having a reciprocal connection with the narrow-field amacrine cell. Thus, for the OFF-bipolar cells, \( e\left( t \right) \) and \( i\left( t \right) \) were provided as

$$ w_{\text{offB}}^{\text{C}} \left( {v_{{\underline{x} ,{\text{C}}}} \left( t \right) - v_{{\underline{x} ,{\text{H}}}} \left( t \right)} \right)^{ + } - w_{\text{offB}}^{\text{offA}} v_{{\underline{x} ,{\text{offA}}}}^{ - } \left( t \right) $$

and \( w_{\text{offB}}^{\text{C}} \left( {v_{{\underline{x} ,{\text{C}}}} \left( t \right) - v_{{\underline{x} ,{\text{H}}}} \left( t \right)} \right)^{ - } - w_{\text{offB}}^{\text{offA}} v_{{\underline{x} ,{\text{offA}}}}^{ + } \left( t \right) \), respectively.

Similarly, for the ON-bipolar cells, \( e\left( t \right) \) and \( i\left( t \right) \) were given as

$$ w_{\text{onB}}^{\text{C}} \left( { - v_{{\underline{x} ,{\text{C}}}} \left( {t - \Delta t_{\text{onB}} } \right) + v_{{\underline{x} ,{\text{H}}}} \left( {t - \Delta t_{\text{onB}} } \right)} \right)^{ + } - w_{\text{onB}}^{\text{onA}} v_{{\underline{x} ,{\text{onA}}}}^{ - } \left( t \right) $$

and \( w_{\text{onB}}^{\text{C}} \left( { - v_{{\underline{x} ,{\text{C}}}} \left( {t - \Delta t_{\text{onB}} } \right) + v_{{\underline{x} ,{\text{H}}}} \left( {t - \Delta t_{\text{onB}} } \right)} \right)^{ - } - w_{\text{onB}}^{\text{onA}} v_{{\underline{x} ,{\text{onA}}}}^{ + } \left( t \right) \), respectively.

Here, the \( \Delta t_{\text{onB}} \) was employed to take into account the signal transmission delay in the ON pathway emerging at the postsynaptic site of ON-bipolar cells (Ashmore and Copenhagen 1980).

1.1.4 Narrow-field amacrine cells

The OFF-type narrow-field amacrine cells were reciprocally connected to the OFF-bipolar cells, thereby enhancing the transient response of the bipolar cells. The inputs from the OFF-bipolar cell to the OFF-type narrow-field amacrine cell, \( e\left( t \right) \) and \( i\left( t \right) \), were provided as \( w_{\text{offA}}^{\text{offB}} v_{{\underline{x} ,{\text{offB}}}}^{ + } \left( t \right) \) and \( w_{\text{offA}}^{\text{offB}} v_{{\underline{x} ,{\text{offB}}}}^{ - } \left( t \right) \), respectively.

Similarly, \( e\left( t \right) \) and \( i\left( t \right) \) for the ON-type narrow-field amacrine cell were provided as \( w_{\text{onA}}^{\text{onB}} v_{{\underline{x} ,{\text{onB}}}}^{ + } \left( t \right) \) and \( w_{\text{onA}}^{\text{onB}} v_{{\underline{x} ,{\text{onB}}}}^{ - } \left( t \right) \), respectively.

1.1.5 Wide-field amacrine cells

Each of the wide-field amacrine cells was assumed to receive a depolarizing input from a pool of seven OFF- and seven ON-bipolar cells. Hence, \( e\left( t \right) \) was provided as

$$ w_{\text{wfA}}^{\text{offB}} \mathop \sum \limits_{{\underline{j} = \underline{x}^{1} }}^{{\underline{x}^{7} }} v_{{\underline{j} ,{\text{offB}}}}^{ + } \left( t \right) + w_{\text{wfA}}^{\text{onB}} \mathop \sum \limits_{{\underline{j} = \underline{x}^{1} }}^{{\underline{x}^{7} }} v_{{\underline{j} ,{\text{onB}}}}^{ + } \left( t \right) . $$

Since, in the present model, there was no hyperpolarizing input, the term \( i\left( t \right) \) was eliminated for the wide-field amacrine cells.

1.1.6 Ganglion cells

Similar to the wide-field amacrine cells, each unit of the ganglion cells was assumed to receive a depolarizing input from a pool of seven OFF- and seven ON-bipolar cells. Hence, \( e\left( t \right) \) was provided as

$$ w_{\text{G}}^{\text{offB}} \mathop \sum \limits_{{\underline{j} = \underline{x}^{1} }}^{{\underline{x}^{7} }} v_{{\underline{j} ,{\text{offB}}}}^{ + } \left( t \right) + w_{\text{G}}^{\text{onB}} \mathop \sum \limits_{{\underline{j} = \underline{x}^{1} }}^{{\underline{x}^{7} }} v_{{\underline{j} ,{\text{onB}}}}^{ + } \left( t \right) . $$

Since, in the present model, the ganglion cell was modeled to not receive inhibitory input from cells in the same column, i(t) was set to zero. The gap junctions within the ganglion cell network were also implemented by the fourth term in Eq. (1) as

$$ \mathop \sum \limits_{{\underline{j} = \underline{x}^{1} }}^{{\underline{x}^{6} }} w_{\text{G}}^{\text{G}} v_{{\underline{j} ,{\text{G}}}} \left( t \right) . $$

Here, \( \underline{j} \) was \( \underline{x}^{1} , \underline{x}^{2} , \ldots , \underline{x}^{6} \) as each ganglion cell was adjacent to six other ganglion cells on the hexagonal grid. Although i(t) was set to zero, the ganglion cells receive inhibitory signals from distant areas via the wide-field amacrine cells, which was governed by the term,

$$ - w_{\text{G}}^{\text{wfA}} \mathop \sum \limits_{{\underline{j} = \underline{x}^{1} }}^{{\underline{x}^{18} }} v_{{\underline{j} ,{\text{wfA}}}} \left( t \right) - w_{\text{G}}^{\text{wfA}} \mathop \sum \limits_{{\underline{j} = \underline{x}^{1} }}^{{\underline{x}^{24} }} v_{{\underline{j} ,{\text{wfA}}}} \left( t \right) . $$

These two terms indicated the inhibitory inputs coming from the 18 amacrine cells located at three units apart and from the 24 cells located four units apart on the hexagonal grid (Fig. 1, wfA axon). This wide-field signaling was also utilized for a mechanism of the contrast gain control (Fig. 1B, box c) as previously suggested (van Hateren et al. 2002; Thiel et al. 2006; Berry et al. 1999). The signal from the wide-field amacrine cell was temporally processed by a low-pass filter with a time constant \( \tau_{c} \) and a constant delay \( \Delta t_{c} \). The processed signal was then fed to a nonlinearity function (Berry et al. 1999; Crevier and Meister 1998),

$$ f\left( v \right) = \frac{1}{{1 + \left( {c_{1} v} \right)^{4} }} $$

and then limited by a simple nonlinearity of

$$ \frac{{c_{2} f\left( v \right)}}{{c_{3} + f\left( v \right)}}, $$

where \( v \) represents the filtered wfA-mediated signal, \( c_{1} \) determines the skewness of the nonlinearity, and \( c_{2} \) and \( c_{3} \) determine a saturation level of the gain control. The generator potential was multiplied by the output of this gain-control cascade (Berry et al. 1999) and a simple thresholding mechanism, which generated a spike once the potential exceeded a certain threshold (\( \theta \)). Note that when the wfA-mediated inhibition and ganglion cell depolarization perfectly match in time, ganglion cell potential is canceled out before it is multiplied by the gain control; therefore, no spike output is produced.

1.2 Model parameter values

See Table 1.

Table 1 Parameter values used in simulations
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Sağlam, M., Hayashida, Y. A single retinal circuit model for multiple computations. Biol Cybern 112, 427–444 (2018). https://doi.org/10.1007/s00422-018-0767-9

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