One of the most fundamental laws of physics is the principle of least action. Motivated by its predictive power, we introduce a neuronal least-action principle for cortical processing of sensory streams to produce appropriate behavioral outputs in real time. The principle postulates that the voltage dynamics of cortical pyramidal neurons prospectively minimizes the local somato-dendritic mismatch error within individual neurons. For output neurons, the principle implies minimizing an instantaneous behavioral error. For deep network neurons, it implies the prospective firing to overcome integration delays and correct for possible output errors right in time. The neuron-specific errors are extracted in the apical dendrites of pyramidal neurons through a cortical microcircuit that tries to explain away the feedback from the periphery, and correct the trajectory on the fly. Any motor output is in a moving equilibrium with the sensory input and the motor feedback during the ongoing sensory-motor transform. Online synaptic plasticity reduces the somatodendritic mismatch error within each cortical neuron and performs gradient descent on the output cost at any moment in time. The neuronal least-action principle offers an axiomatic framework to derive local neuronal and synaptic laws for global real-time computation and learning in the brain.

Experience shapes the brain as neural circuits can be modified by neural stimulation or the lack of it. The molecular mechanisms underlying structural circuit plasticity and how plasticity modifies behaviour are poorly understood. Subjective experience requires dopamine, a neuromodulator that assigns a value to stimuli, and it also controls behaviour, including locomotion, learning, and memory. In Drosophila, Toll receptors are ideally placed to translate experience into structural brain change. Toll-6 is expressed in dopaminergic neurons (DANs), raising the intriguing possibility that Toll-6 could regulate structural plasticity in dopaminergic circuits. Drosophila neurotrophin-2 (DNT-2) is the ligand for Toll-6 and Kek-6, but whether it is required for circuit structural plasticity was unknown. Here, we show that DNT-2-expressing neurons connect with DANs, and they modulate each other. Loss of function for DNT-2 or its receptors Toll-6 and kinase-less Trk-like kek-6 caused DAN and synapse loss, impaired dendrite growth and connectivity, decreased synaptic sites, and caused locomotion deficits. In contrast, over-expressed DNT-2 increased DAN cell number, dendrite complexity, and promoted synaptogenesis. Neuronal activity modified DNT-2, increased synaptogenesis in DNT-2-positive neurons and DANs, and over-expression of DNT-2 did too. Altering the levels of DNT-2 or Toll-6 also modified dopamine-dependent behaviours, including locomotion and long-term memory. To conclude, a feedback loop involving dopamine and DNT-2 highlighted the circuits engaged, and DNT-2 with Toll-6 and Kek-6 induced structural plasticity in this circuit modifying brain function and behaviour.