Biophysical Model-Comparison Notes¶

These notes describe how the Tensor-Field Neural Equation (TFNE) formulations in jaxfne relate to established point-neuron, multicompartment, and extracellular-readout formulations from the computational-neuroscience literature. The goal is to map the package's operator chain onto familiar mathematical building blocks.

1. Operator chain¶

The standard jaxfne workflow maps a multiscale circuit onto a tensor-parallel computation graph:

[Emitter (Local Spiking / HH Dynamics)]
        │
        ▼ (Presynaptic Spikes / Conductances)
[Synapse / Connectivity Layer]
        │
        ▼ (Recurrent Drive & Aggregate Inputs)
[Source Layer (Transmembrane Current Bookkeeping)]
        │
        ▼ (Laminar / Spatial Geometry Weighting)
[Passive Field / Linear Readout]
        │
        ▼ (Extracellular Contact Probes)
[LFP / CSD / EEG / MEG Readout Proxies]

Local nonlinearities (voltage gates, synaptic receptors, adaptation variables) live inside the Emitter and Synapse components, while spatial propagation and electrode readouts are represented as parallelized Linear Readout operators.

2. Core modeling formulations¶

jaxfne adopts canonical mathematical formulations used across computational-neuroscience benchmarks:

2.1. Local conductance & adaptation¶

Multicompartment Hodgkin-Huxley-style gate kinetics, calcium shells, and voltage-reset adaptation are represented inside the generalized Emitter loop.

2.2. Network connectivity & synaptic kernels¶

Point-neuron approximations and compartmental models map recurrent connectivity weights through explicit synaptic trace equations. Postsynaptic current updates follow:

Exponential Synapse: $$s_j[t+1] = s_j[t]\exp\left(-\frac{\Delta t}{\tau_j}\right) + z_j[t]$$
Alpha Synapse: $$k_j(t) = \frac{t}{\tau_j}\exp\left(1-\frac{t}{\tau_j}\right)$$
Double-Exponential Synapse: $$k_j(t) = A_j \left(e^{-t/\tau_{decay,j}} - e^{-t/\tau_{rise,j}}\right)$$

2.3. Dual multiscale representation¶

TFNE architectures separate cellular dynamics from low-dimensional network descriptions (e.g., GLIF vs. full biophysical compartment models), so emitter classes can be swapped while a unified linear readout layer is retained.

2.4. Extracellular signal readouts¶

Transmembrane currents are projected onto extracellular recording probes (e.g., laminar silicon probes) using a linear transfer-resistance proxy model: $$Y_c(t) = \sum_n W_{cn} S_n(t)$$

3. Proxy-readout scope¶

Extracellular metrics (LFP-proxy, CSD-proxy, EEG-proxy, MEG-proxy, EMM-proxy) are numerical proxy readouts for objective functions, optimization, and system-level comparison. They use uncalibrated proxy units under the package truth gates (field_solver_status = "linear_solver", physical_amplitude_calibrated = False). See Limitations and future plans for the centralized scope statement.

4. Canonical coding pattern¶

import jaxfne as jtfne

# 1. Instantiate the bridge (e.g. JaxleyBridge)
bridge = jtfne.bridges.JaxleyBridge(
    model=jaxley_model,
    source_mode="transmembrane_current",
    compartment_axis="last"
)

# 2. Extract transmembrane source bookkeeping (proxy units)
sources = bridge.extract_sources(simulation_result)

# 3. Plot proxy signals using the visualizer namespace
fig = jtfne.vis.lfp(sources)

References¶

Arkhipov et al. (2018), Gouwens et al. (2018), Billeh et al. (2020), Rimehaug et al. (2023) — point-neuron, multicompartment, and extracellular-readout formulations whose mathematical structure the TFNE operator chain mirrors.