Draft of chapter 6, Spike initiation with an initial segment

I have just uploaded an incomplete draft of chapter 6, "Spike initiation with an initial segment". This chapter deals with how spikes are initiated in most vertebrate neurons (and also some invertebrate neurons), where there is a hotspot of excitability close to a large soma. This situation has a number of interesting implications which make spike initiation quite different from the situation investigated by Hodgkin and Huxley, that of stimulating the middle of an axon. Most of the chapter describes the theory that I have developed to analyze this situation, called "resistive coupling theory" because the axonal hotspot is resistively coupled to the soma.

The chapter is currently unfinished, because a few points require a little more research, which we have not finished. The presentation is also a bit more technical than I would like, so this is really a draft. I wanted nonetheless to release it now, as I have not uploaded a chapter for a while and it could be some time before the chapter is finished.

What is computational neuroscience? (XXVIII)The Bayesian brain

Our sensors give us an incomplete, noisy, and indirect information about the world. For example, estimating the location of a sound source is difficult because in natural contexts, the sound of interest is corrupted by other sound sources, reflections, etc. Thus it is not possible to know the position of the source with certainty. The ‘Bayesian coding hypothesis’ (Knill & Pouget, 2014) postulates that the brain represents not the most likely position, but the entire probability distribution of the position. It then uses those distributions to do Bayesian inference, for example, when combining different sources of information (say, auditory and visual). This would allow the brain to optimally infer the most likely position. There is indeed some evidence for optimal inference in psychophysical experiments – although there is also some contradicting evidence (Rahnev & Denison, 2018).

The idea has some appeal. The problem is that, by framing perception as a statistical inference problem, it focuses on the most trivial type of uncertainty, statistical uncertainty. It is illustrated by the following quote: “The fundamental concept behind the Bayesian approach to perceptual computations is that the information provided by a set of sensory data about the world is represented by a conditional probability density function over the set of unknown variables”. Implicit in this representation is a particular model, for which variables are defined. Typically, one model describes a particular experimental situation. For example, the model would describe the distribution of auditory cues associated with the position of the sound source. Another situation would be described by a different model, for example one with two sound sources would require a model with two variables. Or if the listening environment is a room and the size of that room might vary, then we would need a model with the dimensions of the room as variables. In any of these cases where we have identified and fixed parametric sources of variation, then the Bayesian approach works fine, because we are indeed facing a problem of statistical inference. But that framework doesn’t fit any real life situation. In real life, perceptual scenes have variable structure, which corresponds to the model in statistical inference (there is one source, or two sources, we are in a room, the second source comes from the window, etc). The perceptual problem is therefore not just to infer the parameters of the model (dimensions of the room etc), but also the model itself, its structure. Thus, it is not possible in general to represent an auditory scene by a probability distribution on a set of parameters, because the very notion of a parameter already assumes that the structure of the scene is known and fixed.

Inferring parameters for a known statistical model is relatively easy. What is really difficult, and is still challenging for machine learning algorithms today, is to identify the structure of a perceptual scene, what constitutes an object (object formation), how objects are related to each other (scene analysis). These fundamental perceptual processes do not exist in the Bayesian brain. This touches on two very different types of uncertainty: statistical uncertainty, variations that can be interpreted and expected in the framework of a model; and epistemic uncertainty,  the model is unknown (the difference has been famously explained by Donald Rumsfeld).

Thus, the “Bayesian brain” idea addresses an interesting problem (statistical inference), but it trivializes the problem of perception, by missing the fact that the real challenge is epistemic uncertainty (building a perceptual model), not statistical uncertainty (tuning the parameters): the world is not noisy, it is complex.