Dynamical Systems of the BCM Learning Rule: Emergent Properties and Application to Clustering

The BCM learning rule has been used extensively to model how neurons in the brain cortex respond to stimulus. One reason for the popularity of the BCM learning rule is that, unlike its predecessors which use static thresholds to modulate neuronal activity, the BCM learning rule incorporates a dynamic threshold that serves as a homeostasis mechanism, thereby providing a larger regime of stability. This dissertation explores the properties of the BCM learning rule – as a dynamical system– in different time-scale parametric regimes. The main observation is that, under certain stimulus conditions, when homeostasis is at least as fast as synapse, the dynamical system undergoes bifurcations and may trade stability for oscillations, torus dynamics, and chaos. Analytically, it is shown that the conditions for stability are a function of the homeostasis time-scale parameter and the angle between the stimuli coming into the neuron. When the learning rule achieves stability, the BCM neuron becomes selective. This means that it exhibits high-response activities to certain stimuli and very low-response activities to others. With data points as stimuli, this dissertation shows how this property of the BCM learning rule can be used to perform data clustering analysis. The advantages and limitations of this approach are discussed, in comparison to a few other clustering algorithms

Oscillations and chaos in the dynamics of the BCM learning rule

The BCM learning rule originally arose from experiments intended for measuring the selectivity of neurons in the primary visual cortex, and it dependence on input stimuli. This learning rule incorporates a dynamic LTP threshold, which depends on the time averaged postsynaptic activity. Although the BCM learning rule has been well studied and some experimental evidence of neuronal adherence has been found in the other areas of the brain, including the hippocampus, there is still much to be known about the dynamic behavior of this learning rule

Neuromatch Academy: a 3-week, online summer school in computational neuroscience

Neuromatch Academy (https://academy.neuromatch.io; (van Viegen et al., 2021)) was designed as an online summer school to cover the basics of computational neuroscience in three weeks. The materials cover dominant and emerging computational neuroscience tools, how they complement one another, and specifically focus on how they can help us to better understand how the brain functions. An original component of the materials is its focus on modeling choices, i.e. how do we choose the right approach, how do we build models, and how can we evaluate models to determine if they provide real (meaningful) insight. This meta-modeling component of the instructional materials asks what questions can be answered by different techniques, and how to apply them meaningfully to get insight about brain function