The Lagrangian Brain: A Constraint Mechanics of Mind and Machine

Yuhang He, Si Wu

A brain is in one state at a time, yet most of what decides its next state is absent from it: the patterns its world never produces, the statistics of a lifetime, a goal that lies ahead. In mechanics a wire acts on a bead only through a force that appears when the bead presses on it, and Lagrange computed such forces with one multiplier for each constraint. We propose that a brain follows the same mechanics. Everything a circuit knows, is told or wants is a constraint on its single state and acts through its own multiplier, which accumulates the violation and pushes the state back. Neurons and synapses carry the multipliers, each with its own time constant: inhibitory interneurons carry what never happens, adaptation the cost of staying in one state, synapses the statistics of experience, and feedback to apical dendrites the future. From this hypothesis follow Dale's law, the reciprocal wiring and balance of excitation and inhibition, the paradoxical responses of cortex and the frequencies of brain rhythms. Belief becomes the time one trajectory spends in each state, learning becomes sleep proposing laws and waking refuting the false ones, and planning becomes a wave from the…