In the human brain, "learning" is a very broad term. It takes many forms, from classical and operant conditioning to one-shot memorization of faces and even in some cases eidetic memory for mathematical equations or chess boards or the surfaces of musical instruments. Our brains use all of these forms, and all of the transmission forms discussed earlier. However learning begins well before birth. Axons find their way to their targets. Synapses form, then they get pruned and some of them disappear. There are "critical periods" when connections may form quickly and with the assistance of genetically programmed neural activity, for example the retinal waves that contribute to the development of the superior colliculus, and the mapping of the auditory pathway from the inferior colliculus over the retinotopic map in the superior colliculus (the alignment of this map is essential for orienting and navigation). In many cases the intuition of what constitues an "error" is straightforward. If I show you a dog and you tell me it's a cat, that's an error. And, if a topographic axon somehow connects with an inappropriate target, that's an error too. But what about situations like the theta wave in the hippocampus, or the retinal waves during visual development? Are there "errors" in these phenomena, that might need to be accounted for? The retinotopic map from the eye to the superior colliculus depends on retinal waves to form properly, and disruption of the waves prevents the proper alignment of not only this map, but also the subsequent auditory map from the inferior colliculus. Apparently in this case the "error" in the mapping is being ignored (or not processed), because the map forms anyway but it's off center and distorted.

Since we've already talked about retinal waves in previous sections, let's use them as an example. These waves begin long before birth, even before rod and cone maturation and before functional vision is possible. In humans, the retina forms from the anterior neural plate. Around 3-4 weeks the optic cup differentiates into two layers, the inner layer becomes the retina and the outer layers becomes the pigment epithelium that will abut the photoreceptors. By 8 weeks the neural retina already contains photoreceptors, bipolar cells, and ganglion cells, and the retinal layers are in place. The inner and outer synaptic layers are maturing around 16 weeks. They are due to the activity of a particular kind of amacrine cell, called starburst amacrine. They begin at the periphery of the retina, around the outer boundary, and move inward as spherical waves. This activity induces direction sensitivity in the plastic synapses feeding the retinal ganglion cells. The direction of these cells ends up aligned with the "optic flow" that the organism experiences when it's moving forward in its environment. This flow is pushed forward as directional sensitivity in the superior colliculus.

Retinal waves can be visualized with calcium imaging and multi-electrode arrays. They can be seen engaging both amacrine and ganglion cells. Waves continue during adulthood, but they acquire a different character. They become involved in things like the retinal shift effect, which is mediated by amacrine cells. The psychologist Alberta Gilinsky studied several forms of perceptual alterations caused in the retina itself, including judgements of size and distance. Technically these could be considered "errors" of a sort, but they're not treated this way by our visual systems - rather they end up contributing to "illusions", and subtle changes in perception that sometimes go barely noticed.

One of the downstream dependencies of retinal waves is the looming reflex, which is organized in the superior colliculus, and depends on the directional sensitivity of the optic flow generated by the retina. The looming reflex is especially prominent in rodents, that are prey for large birds flying overhead. A dark shape presented in the upper periphery of a rodent retina will cause an avoidance reaction to the opposite side. This reaction is mediated by the superior colliculus, which receives retinotopic input from peripheral magnocellular neurons in the retina that are sensitive to both motion and direction. In humans this reflex is expanded to include objects that appear to be moving rapidly towards oneself, and it usually includes a startle or flinch, some eye movements, and autonomic changes like increased heart rate or pupil dilation, as well as an orienting (avoidance) reaction (see Thieu et al 2024).

"Errors" in the looming reflex can be deadly. In this case, the connectivity between the retina and the superior colliculus is genetically patterned and undergoes a critical period even before birth. The optic flow experienced during adulthood is set in place well before the visual systems are mature, by mechanisms that have somehow arisen during evolution (clearly, avoidance of predators is important for the survival of the organism!).

Optimization

In an optimization scenario, we endeavor to minimze an energy function, which is also called a cost function or an error function. An energy function is usually defined theoretically, whereas a cost function is defined relative to a resource, and an error function is defined relative to a correct response (a "standard"). In certain cases these three things can be one and the same, but mostly we should pay attention to their specific meanings. Many energy functions are contrived, and don't relate in a natural way to the represented parameters in neurons and synapses. And we've already discussed the non-biological nature of back propagation, where errors are sent backwards through the network layers between stimulus presentations.

We can separate an optimization effort into sequential stages. In the first stage, the state of the network is determined. In the second stage, it is compared against a standard. In the third stage, an error signal is derived. And in the final stage, the error signal is applied to the state of the network. In the case of a Hamiltonian, the error signal may be a single number (representing the aggregate energy in the entire network), however in the general case the error signal is a "tensor field" where every point in the network has a (possibly multi-dimensional) error vector.

The computational complexity of an optimization effort depends heavily on the types of neurons and synapses in the network. In most cases, optimizing 10,000 neurons or more requires that we sacrifice some detail in the underlying differential equations, in favor of computational tractability and performance. In many machine learning experiments, pre-processing (like vectorization) is not counted as part of the performance cost, but biological realism won't let us get away with that kind of handwaving. In regards to realistic networks, one of the important behaviors to understand is optimization with spiking neurons, especially when the input has been pre-processed with phase encoding.

Recovering Spike Trains From Phase-Encoded Signals

The phase encoding of spatial and temporal information in the hippocampal region does not encode absolute time. It only encodes the sequence of events. How does one recover precise spike timing information upon playback? The short answer is, one doesn't. Precision in spike timing comes from somewhere else. The purpose of precise spike timing is to match environmental requirements, and so the playback is variable depending on the timing of events. However one of the interesting thing about the phase encoding is sequences can be played back both forwards and backwards, quickly or slowly, while still retaining proper internal ordering.

In general there are four methods for recovering spike timing from a phase encoded signal. If you know the original encoding frequency and phase, you can use it to recover spike times, but unfortunately the original frequency and phase are probably not reproducible in the stochastic and noisy brain. You can use an inverse Fourier transform, which is a mathematical procedure that would require specific brain wiring, and there is no evidence for such a thing at this time. You can cross-correlate with a known reference oscillation, and this is a usable method but it only provides estimates, not ground truth. And finally, there is spike train resampling, which would necessarily use interpolation or a delay line mechanism to recover precise timing. Somewhere between methods 3 and 4 is where the brain ends up, it's very good at sampling and correlation. For modeling purposes, tools like FieldTrip and Elephant are available for spike train processing, analysis, and resampling.