Figures and data
![](https://prod--epp.elifesciences.org/iiif/2/89674%2Fv2%2Fcontent%2F534198v3_fig1.tif/full/max/0/default.jpg)
Somato-dendritic mismatch energies and the neuronal least-action (NLA) principle.
(a1) Sketch of a cross-cortical network of pyramidal neurons described by NLA. (a2) Correspondence between elements of NLA and biological observables such as membrane voltages and synaptic weights. (b1) The NLA principle postulates that small variations δũ (dashed) of the trajectories ũ (solid) leave the action invariant, δA = 0. It is formulated in the look-ahead coordinates ũ (symbolized by the spyglass) in which ‘hills’ of the Lagrangian (shaded grey zones) are foreseen by the prospective voltage so that the trajectory can turn by early enough to surround them. (b2) In the absence of output nudging (β = 0), the trajectory u(t) is solely driven by the sensory input, and prediction errors and energies vanish (L = 0, outer blue trajectory at bottom). When nudging the output neurons towards a target voltage (β > 0), somato-dendritic prediction errors appear, the energy increases (red dashed arrows symbolising the growing ‘volcano’) and the trajectory u(t) moves out of the L = 0 hyperplane, riding on top of the ‘volcano’ (red trajectory). Synaptic plasticity
![](https://prod--epp.elifesciences.org/iiif/2/89674%2Fv2%2Fcontent%2F534198v3_fig2.tif/full/max/0/default.jpg)
Prospective coding in cortical pyramidal neurons enables instantaneous voltage-to-voltage transfer.
(a1) The instantaneous spike rate of cortical pyramidal neurons (top) in response to sinusoidally modulated noisy input current (bottom) is phase-advanced with respect to the input (adapted from Köndgen et al., 2008). (a2) Similiarly, in NLA, the instantaneous firing rate of a model neuron
![](https://prod--epp.elifesciences.org/iiif/2/89674%2Fv2%2Fcontent%2F534198v3_fig3.tif/full/max/0/default.jpg)
Moving equilibrium hypothesis for motor control and real-time learning of cortical activity.
(a) A voluntary movement trajectory can be specified by the target length of the muscles in time,
![](https://prod--epp.elifesciences.org/iiif/2/89674%2Fv2%2Fcontent%2F534198v3_fig4.tif/full/max/0/default.jpg)
On-the-fly learning of finger responses to visual input with real-time Dendritic Error Propagation (rt-DeEP).
(a) Functionally feedforward network with handwritten digits as visual input
![](https://prod--epp.elifesciences.org/iiif/2/89674%2Fv2%2Fcontent%2F534198v3_fig5.tif/full/max/0/default.jpg)
Hierarchical plastic microcircuits implement real-time Dendritic Error Learning (rt-DeEL).
(a) Microcircuit with ‘top-down’ input (originating from peripheral motor activity, blue line) that is explained away by the lateral input via interneurons (dark red), with the remaining activity representing the error ēl. Plastic connections are denoted with a small red arrow and nudging with a dashed line. (b1) Simulated network with 784-300-10 pyramidal-neurons and a population of 40 interneurons in the hidden layer used for the MNIST learning task where the handwritten digits have to be associated to the 10 fingers. (b2) Test errors for rt-DeEL with joint tabula rasa learning of the forward and lateral weights of the microcircuit. A similar performance is reached as with classical error backpropagation. For comparability, we also show the performance of a shallow network (dashed line). (b3) Angle derived from the Frobenius norm between the lateral pathway
![](https://prod--epp.elifesciences.org/iiif/2/89674%2Fv2%2Fcontent%2F534198v3_fig6.tif/full/max/0/default.jpg)
Recovering presynaptic potentials through short term depression.
(a1) Relative voltage response of a depressing cortical synapse (recreated from Abbott et al., 1997), identified as synaptic release probability p. (a2) The product of the low-pass filtered presynaptic firing rate
![](https://prod--epp.elifesciences.org/iiif/2/89674%2Fv2%2Fcontent%2F534198v3_utbl1.tif/full/max/0/default.jpg)
![](https://prod--epp.elifesciences.org/iiif/2/89674%2Fv2%2Fcontent%2F534198v3_utbl1a.tif/full/max/0/default.jpg)