Axial Diffusion

[thorstenhater]

Introduction

Plasticity processes are mediated by signalling ions, eg Ca++, which are generated by
synapses upon reception of a spike. These ions then propagate along the dendrite
by diffusion and are received by other synapses, modifying their weight upon reception.

We introduce a simplified -- as opposed to solving the Nernst-Planck equation -- approach
to model these processes. Concretely, we will introduce a new field, dubbed Xd, to the
NMODL dynamics, with the following properties

• Xd can be read from and written to by NMODL density and point mechanisms.
  This allows us to model decay, consumption, and sources, as well as interaction with the actual processes.
• Xd propagates according to a diffusion law ∂_t Xd = ∂_z 1/rX ∂_z Xd + iX/qi along the neuron, where
  the per-species resistivity rX may vary in space (accessible via the paint-interface) but not in time.
• NOTE: Currently this follows a simplified update model, where Xd is atomically updated during the
  advance_state phase and not like other ionic quantities in two steps (once as a STATE and later effects
  are applied in write_ions / compute_currents.)
  • We could get away with this as there is no precedent in NRN and we can make our own semantics to fit our needs
  • However diverting from the ionic model feels gross.

Update Nov 2021

After the initial exploration and design a number of issues became apparent.

1. Arbor (and Neuron) model the neuron as a thin cable in the following limit case

• External and internal concentrations are constant in time; as they are linked to an infinite buffer of the relevant species.
• Currents iX across the membrane do not contribute to Xi/Xo.
• Similarly, axial currents to not change Xi.
• Neither are ions transported by concentration gradients.
• Therefore, Xi and Xo are reset every dt and we cannot use Xi for our model.

2. Partial presence of a diffusing species is non-sensical

• Basically, the matrix solver requires a defined value everywhere

3. All dynamics are only well-defined in proximity of the membrane, essentially ignoring the interior of the neuron.
4. Concentration models write to a state variable shadowing Xi

Changes

• Add paint and default interfaces.

  • Diffusivity is paintable on regions like init_ext_conc
  • ... and can be added as a default via set_ion (Python) or set_default (C++)
  • see python/example/diffusion.py

• Add per-ion conductivity gX, needed for matrix solver

  • part of the ABI, got bumped to 0.1.1
  • modcc computes and accumulates gX on compute_currents iff iX is written
  • shared_state sets up the backing store

• Add per-ion diffusion solvers

• Clean-up

  • Remove redundant arrays
  • Remove matrix.hpp
  • Push matrix_state into shared_state

Open Questions/Issues

• (How to) disable ion-conductivity if we do not need it?
• Currently the cable equation and diffusion solvers (1 + 1/diffusing ion) store duplicates of the solver state (eg parent indices), it would be nice to have one copy in total.
• What to do if diffusivity defaults to a finite value, but is overridden to be zero in a region?

References

Linked Issues

Closes #1651

[brenthuisman]

bors try

[bors]

try

Build failed:

• ci/gitlab/gitlab.com

[thorstenhater]

Again an Ault fail state, Mr Bors

[AdhocMan]

I've only found a few small issues

[AdhocMan]

Looks good from my side.