We present a new class of accurate, efficient, and scalable models for the activity patterns of large neural populations. These models rely on sparse, random, and nonlinear projections and are highly accurate in representing the codebook of populations of hundreds of neurons, requiring surprisingly small amounts of training data, while also being scalable and computationally efficient. Interestingly, these models have a biologically plausible implementation via shallow, simple neural circuits that use random, sparse, and nonlinear projections, and can learn using a simple, noise-driven learning rule. We further show that homeostatic synaptic scaling enhances both the efficiency and accuracy of learning such models for very large neural populations. Finally, we discuss how these models may enable the brain to perform Bayesian decoding and to learn metrics over the space of neural codes and external stimuli.

 I will describe results showing that neural responses in the hippocampus have a low-dimensional hyperbolic geometry and that their hyperbolic size is optimized for the number of available neurons. It was also possible to analyze how neural representations change with experience. In particular, neural representations continued to be described by a low-dimensional hyperbolic geometry as the animal explored the environment but the radius increased logarithmically with time. This time dependence matches the maximal rate of information acquisition by a maximum entropy discrete Poisson process, further implying that neural representations continue to perform optimally as they change with experience. I will also discuss how and why these results generalize to other levels of biological organization including viral evolution and cell differentiation processes.

Dendritic spines carry thousands of synaptic spines and every spine harbours hundreds of protein species. Hereby, the necessary number of proteins must be maintained at all spines and times to keep synaptic homeostasis and to provide the molecular basis for synaptic plasticity. At the same time, the task to distribute molecules with limited lifespans along the widespread and complex dendritic morphologies is highly demanding. Today, it is not clear which rules determine how neurons approach this task. Here, we provide theoretical and experimental evidence showing that energy minimisation is a strong candidate for such a rule. To this end we use a diffusion reaction model to simulate the distribution of mRNAs and proteins at the soma, dendrites and spines. Our model allows us to calculate neuronal mRNA and protein numbers and to predict which mRNA species are more likely to localize in dendrites based on energetic efficiency. Our model predictions are supported by the experimentally measured neuronal translatomes and proteomes comprising tens of thousands of molecular species.