Abstract:Connectomics is a sub-field of Neuroscience aimed at determining connectomes -exact structures of neurons and their synaptic connections in nervous systems. A number of ongoing initiatives at the present time are working towards the goal of ascertaining the connectomes or parts thereof of various organisms. Determining the detailed physiological response properties of all the neurons in these connectomes is out of reach of current experimental technology. It is therefore unclear, to what extent knowledge of th… Show more
“…(This neuron is adapted from [JDS97, ADH07].) b) Studying the connections between neurons has been a subject of inquiry [RB14,Ram19]. Many neural circuits can also be represented by a tree.…”
Any function can be constructed using a hierarchy of simpler functions through compositions. Such a hierarchy can be characterized by a binary rooted tree. Each node of this tree is associated with a function which takes as inputs two numbers from its children and produces one output. Since thinking about functions in terms of computation graphs is getting popular we may want to know which functions can be implemented on a given tree. Here, we describe a set of necessary constraints in the form of a system of non-linear partial differential equations that must be satisfied. Moreover, we prove that these conditions are sufficient in both contexts of analytic and bit-valued functions. In the latter case, we explicitly enumerate discrete functions and observe that there are relatively few. Our point of view allows us to compare different neural network architectures in regard to their function spaces. Our work connects the structure of computation graphs with the functions they can implement and has potential applications to neuroscience and computer science. 2 R. FARHOODI, K. FILOM, I. JONES, K. KORDING 7. Application to neural networks 42 7.1. From neural networks to trees 42 7.2. Trees with repeated labels; a toy example 43 7.3. Trees with repeated labels; estimates 46 References 49
“…(This neuron is adapted from [JDS97, ADH07].) b) Studying the connections between neurons has been a subject of inquiry [RB14,Ram19]. Many neural circuits can also be represented by a tree.…”
Any function can be constructed using a hierarchy of simpler functions through compositions. Such a hierarchy can be characterized by a binary rooted tree. Each node of this tree is associated with a function which takes as inputs two numbers from its children and produces one output. Since thinking about functions in terms of computation graphs is getting popular we may want to know which functions can be implemented on a given tree. Here, we describe a set of necessary constraints in the form of a system of non-linear partial differential equations that must be satisfied. Moreover, we prove that these conditions are sufficient in both contexts of analytic and bit-valued functions. In the latter case, we explicitly enumerate discrete functions and observe that there are relatively few. Our point of view allows us to compare different neural network architectures in regard to their function spaces. Our work connects the structure of computation graphs with the functions they can implement and has potential applications to neuroscience and computer science. 2 R. FARHOODI, K. FILOM, I. JONES, K. KORDING 7. Application to neural networks 42 7.1. From neural networks to trees 42 7.2. Trees with repeated labels; a toy example 43 7.3. Trees with repeated labels; estimates 46 References 49
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