The role of nonlinearities

Sensory adaptation is closely related to the nonlinearities of the pathway it occurs in.

The response curves in this study are a way to measure these nonlinearities. In the studies of divisive gain change and selective coding in AN2 of crickets (Chapters 5 and 6) the change of these nonlinearities was used to quantify adaptation and to test pre-dictions about its functional role. In the theory of Chapter 4 they were used to optimize information transfer. A classic model of sensory pathways is a serial combination of filters and static nonlinearities (Dayan and Abbott, 2001). As discussed above, if the stimulus changes quickly with respect to adaptation time constants, the effect of ad-aptation can be seen as static, resulting in different static nonlinearities. In this case, adaptation can be described by the changes of nonlinearities in the sensory pathway.

One of the central results of this work is that adaptation usually acts on several dis-tinct places along a sensory pathway. What is the effect of the serial combination of adaptation mechanism from the perspective of static nonlinearities?

The kind of change of a single nonlinearity that can be realized is limited by the underlying adaptation mechanism. The most common alterations caused by adapta-tion are additive shifts, for which the nonlinearity is shifted along the abscissa and a scaling, where the entire nonlinearity is multiplied by a factor, resulting in ‘output gain-control’ (cf. discussion of Chapter 5, sect. 5.4.3). By combining several nonlin-earities in a serial manner, more complicated reshaping of the input-output relation of the entire pathway can be realized. An extensive consideration of this is beyond this discussion, but since it constitutes a central design principle behind the organization of adaptation mechanism, a short outlook about the combination of adaptation in two serial nonlinearities (Fig. 7.1) is given in the following.

If the first nonlinearity is shifted and not the second one (Fig. 7.2A1&A2), the result is a shift of the combined nonlinearity (Fig. 7.2A3). Examples for this scenario could be the AN2 of crickets (Chapter 5) at the low frequency carrier or the peripheral adap-tation in the numerical simulations of Chapter 4. If the second nonlinearity shifts but the first one remains unchanged by adaptation, the net effect can be a combination of a

7.3 Design principles for the placement of adaptation

input 1 output 1 = input 2 output 2

nonlinearity 1 nonlinearity 2

stimulus response

Figure 7.1:Sensory pathway as a combination of serial nonlinearities. The stimulus provides input to nonlinearity 1, whose output is the input of the next nonlinearity.

The combined input-output function that describes the mapping of the sensory input to some measure of the final spike output (typically the spike frequency or the spike rate) is really a combination of several - adapting - nonlinearities along the pathway.

shift and a divisive scaling on the output of the combined nonlinearity (Fig. 7.2B1-B3).

This is quite notable, because it suggests that the correct combination of nonlinearities that change subtraction due to adaptation can result in a divisive scaling. In the present work, no obvious example of such an interaction was observed, but the combination of subtractive effects of short term synaptic plasticity and an additional subtractive mod-ulations of the input-output relation has recently been shown to generate divisive gain modulation (Rothman et al., 2009).

A third scenario, which has already been discussed in the context of gain modulation by presynaptic inhibition in Chapter 5, is the combination of a divisive operation and a second nonlinearity (Fig. 7.2C1-C3). Firstly, gain control is preserved when passing through the second nonlinearity. Notably, the output nonlinearity in Fig. 7.2C1 is trans-formed into an input nonlinearity Fig. 7.2C3, which was observed in the high frequency coding of AN2 in crickets (Chapter 5). If the first nonlinearity (Fig. 7.2C1) is character-ized by very steep curves, the saturation of the second nonlinearity limits the response to high-intensity stimuli. If the first nonlinearity becomes shallower, due to adaptation, larger parts of it will be transferred by the second nonlinearity, resulting in a net input gain-control. This effect, even more than the ones shown in Fig. 7.2A&B depends on shape and relative position of the two nonlinearities. An extensive study could be of great interest, but goes beyond the scope of this work. The study of the combination of two nonlinearities affected by adaptation can be extended to longer serial chains of nonlinearities by summarizing the first two nonlinearities in the above way and then combining the resulting nonlinearity with the next one in the chain. For example the output gain-control as a result of the shift in the second nonlinearity (Fig. 7.2B3) could then be seen as the first, divisively affected nonlinearity in Fig. 7.2C1-C3 and it could be possible to generate input gain-control without a real divisively acting adaptation mechanism.

These consideration lead to an additional important design principle for adaptation mechanisms in a sensory pathway: adaptation, when it can be considered to be static

nonlinearity I

Figure 7.2: Effect of adaptation in a pathway with two serial nonlinearities. (A-C) show three different scenarios where and how adaptation affects the combined re-sponse curve of two serial nonlinearities. In (A1-A3), only the first nonlinearity is shifted by adaptation (A1, different colours correspond to different adaptation levels), while the second one remains unchanged (A2). The combined response curve is de-picted in (A3). in (B1-B3), there is a shift in the second nonlinearity (B2) as a result of adaptation. In (C1-C3), adaptation acts divisive on the first nonlinearity (C1). In all cases, the first nonlinearity is modelled by a rectified exponential, the second is gener-ated by a tangent hyperbolic. The dashed lines in (A2, B2 & C2) depict the input range [0-1] that results from the output of nonlinearity 1.

7.3 Design principles for the placement of adaptation relative to the stimulus dynamics, acts primarily on the nonlinearities of the system and identification of these is an important aspect in the study of adaptation.

The goal of this thesis was to ask for general design principles that govern how adapta-tion mechanisms are organized in sensory pathways. In summary, it can be concluded that if one wants to understand the organization of adaptation in a sensory pathway, it is of central importance to understand (1) the time scale, on which adaptation acts relativeto the relevant stimulus aspects, (2) the convergence and divergence of infor-mation, and finally (3) identify the nonlinearities, on which adaptation acts.

Above all, however, the oganization of adpaptation within a sensory pathway cannot be understood without the identification of the stimulus aspects that are relevant for the different parts of that pathway and whose processing is effected by adaptation.

Appendix A