Why nonlinearities?

Studying whether neurons respond to light in a linear or nonlinear way is investigated at two stages in a neuron: at its output and at its inputs. Traditionally, the nonlinearity at the output is examined. A common way to measure the nonlinear output function in visual neurons is by setting a reference light level (e.g. gray) from which the contrast is increased (e.g. to white) or decreased (e.g. to black) by equal amounts (Fig. 1.2A). The response of the neuron is measured (e.g. membrane potential) to the reference level and compared to the increases and decreases in contrast. If a neuron increases its membrane potential to one contrast (e.g. +3mV to white) and decreases the voltage to the opposite contrast by the same amplitude (e.g. -3mV to black), the cell is termed linear (Fig.1.2A). If however, the neuron increases and decreases the membrane

potential with different amounts (0mV to black and +3mV to white) the cell is termed nonlinear.

Often, bipolar cells are approximated by a linear response to light. The view is reinforced by the notion that bipolar cells, similar to photoreceptors and horizontal cells, are non-spiking neurons that respond to light with graded potentials. Retinal ganglion cells, and some amacrine cells, on the other hand, are spiking neurons. Here, the response to light is often approximated

nonlinearly. For example, weak light inputs give rise to no response and only if the input stimulus

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passes a threshold, the neuron spikes. Further, retinal ganglion cells might show response saturation, for example, at a certain point, a further increase in the input magnitude does not increase the spiking rate any further. The described linear and nonlinear responses to light are summarized under the terms stimulus-response transformation, output function or output nonlinearity². All terms are used in this thesis as synonyms.

Another way to measure whether neurons are linear or nonlinear is by studying the spatial integration. A neuron typically receives inputs from multiple upstream neurons. How a neuron combines these inputs into an output is described by its spatial integration property. Thus, here, space refers to light input signals at different spatial locations that activate different presynaptic neurons (Fig.1.2B). Measuring the spatial integration is not as straightforward as for the stimulus-response function. By a set of groundbreaking experiments, Enroth-Cugell and Robson (1966) presented a dark half and a light half inside the receptive field of retinal ganglion cells (see Fig.1.2B for a simplified version of the stimulus). The idea behind the experiment was to assess spatial integration by presenting both positive (e.g. white contrast +1) and negative (e.g. black contrast -1) activation inside the receptive field and study whether the activations with opposite signs can cancel out the response (with the logic of -1+1=0) or not (-1+1>0). They found that some retinal ganglion cells indeed remained silent when presenting such a stimulus and

concluded that for those cells the summation of the presynaptic inputs was approximately done linearly (-1+1=0, Fig.1.2B.) Curiously, they also found retinal ganglion cells that clearly

responded to such light combinations, thus, here the summation over the presynaptic inputs was nonlinear (-1+1>0, Fig.1.2B). The described linear and nonlinear responses are studied under the terms spatial integration, input nonlinearity or spatial nonlinearity³. All terms are used as synonyms.

The stimulus-response transformation and spatial integration are studied by separate stimulus designs. Thus, for each neuron two main types of nonlinearities are generally distinguished in the retina: the output nonlinearity and spatial nonlinearity (Fig.1.2).

2 The term output nonlinearity to describe a linear transformation might be counterintuitive. It has its origin from spiking neurons that have internal nonlinear processes like a spike threshold or a saturation. In this thesis, we will always specify whether the measured output nonlinearity was linear or nonlinear.

3 As for the output nonlinearity, the term spatial or input nonlinearity might be counterintuitive to describe a linear integration. In this thesis, we will always specify whether the spatial integration was linear or nonlinear.

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Figure 1.2. The two types of nonlinearities. A. Output nonlinearity, also termed stimulus-response

transformation, here the light signal (input) is related to the neuronal response (output). A reference light level is set (e.g. gray) from which the contrast is increased (e.g. white, in Weber contrast +1) and decreased (e.g. black, in Weber contrast -1). The neuron’s output is measured in millivolts. A linear cell responds with equal amount of negative and positive deflection from the voltage at the reference light level. A nonlinear cell responds with different amounts. B. Spatial nonlinearity, also termed spatial integration, here the input neurons are stimulated with dark (-1, in Weber contrast) and bright (+1, in Weber contrast) contrast. If the activation with opposite sign cancels out the response (-1+1=0), the cell is termed linear, if however the cell responds (-1+1≠0) it is termed nonlinear.

It has been proposed that at the heart of the feature extraction, and thus response diversity in retinal ganglion cells, lie nonlinear signal transformations from bipolar cells to ganglion cells (Gollisch, 2013; Gollisch and Meister, 2010; Roska and Meister, 2014). For example, nonlinear spatial integration in retinal ganglion cells cannot be explained by a linear signal transmission. Further, linear signal transmission to ganglion cells cannot explain the response to small objects that move differently from the background detected by object-motion-sensitive ganglion cells originally termed “bug perceiver” (Baccus et al., 2008; Lettvin et al., 1959;

Ölveczky et al., 2003; Zhang et al., 2012) or the sensitivity of some ganglion cells to approaching objects, also termed “approach-sensitive” or “looming detectors” (Munch et al., 2009).

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Yet, studies measuring the stimulus-response relationship in bipolar cells’ somas show controversial results. Certain studies show a linear stimulus-response relationship (Baccus and Meister, 2002; Dacey et al., 2000; Fahey and Burkhardt, 2003; Rieke, 2001; Sakai and Naka, 1987a; Toyoda, 1974) and other studies reported nonlinear bipolar cell responses (Burkhardt and Fahey, 1998; Euler and Masland, 2000; Fahey and Burkhardt, 2003). It has been speculated that the inconsistency between the different studies is due to different stimulus dynamics and bipolar cell types (Burkhardt and Fahey, 1998; Schwartz and Rieke, 2011). Yet, whether different stimuli produce different nonlinear properties, as well as whether different bipolar cell types (e.g.

sustained vs. transient) show different nonlinearities, is not understood. Moreover, how the particular form of the nonlinear stimulus-response relationship in bipolar cells looks like is unclear. Traditionally in computational models, they are approximated by a threshold-linear transformation (Gollisch, 2013; Gollisch and Meister, 2010), yet recently also threshold-quadratic transformations were proposed (Bolinger and Gollisch, 2012).

In addition, how bipolar cells themselves integrate their inputs from presynaptic photoreceptors and horizontal cells is not known. Yet, the spatial integration in bipolar cells is broadly assumed to occur linearly. However, evidence for nonlinear neurotransmitter release in photoreceptors, is challenging the view of a linear integration in bipolar cells. For example nonlinear signals in rods (Dunn and Rieke, 2008; Field and Rieke, 2002; van Rossum and Smith, 1998) and in cones (Baden et al., 2013c; Dunn et al., 2007) have been reported.