1. The problem from first principles
A nervous system is a physical system. Its state at time t includes:
| State variable | Physics | Timescale |
|---|---|---|
| Membrane potential along every neurite | Cable equation (charge conservation) | 10 µs – 10 ms |
| Ion channel gating states | Markov / Hodgkin–Huxley kinetics | 0.1 ms – 1 s |
| Ion concentrations (Ca²⁺, K⁺, Na⁺, Cl⁻), in and out of cells | Reaction–diffusion, pumps, buffers | 1 ms – minutes |
| Vesicle pools, release machinery, receptor states | Stochastic kinetics | 1 ms – seconds |
| Neuromodulators in extracellular space | Diffusion + uptake/degradation | 100 ms – minutes |
| Second messengers (cAMP, IP₃, DAG, Ca²⁺ stores) | Biochemical networks | 100 ms – minutes |
| Channel/receptor expression levels | Activity-dependent gene regulation | minutes – days |
| Body, muscles, sensors, environment | Continuum and rigid-body mechanics, fluids, optics, chemistry | 1 ms – seconds |
The dynamics are dx/dt = F(x; G, θ, u), where:
- G is structure. Some of it is measured by EM: neuron morphology, chemical synapse locations, synapse sizes and neurotransmitter predictions.
- θ is cellular and molecular parameters: channel densities, receptor types and kinetics, release probabilities, gap-junction conductances, neuromodulator release and receptor maps. Almost none of this is measured per neuron.
- u is sensory input. It is not external. It is produced by the body moving through the environment, and the movement is produced by the nervous system.
The core difficulty
The connectome gives you most of G. It gives you almost none of θ.
| C. elegans | Adult Drosophila CNS | |
|---|---|---|
| Neurons | 302 | ~140k (FlyWire brain) to ~166.7k (male CNS, 2026) |
| Chemical synapses | ~7k connections | ~50M (brain) to ~125M (CNS) |
| Unknown parameters if fitted per neuron and per compartment | ~10⁵–10⁶ | ~10⁹–10¹⁰ |
| Neurons recordable at once with whole-brain imaging | ~all head neurons (calcium only) | a small fraction at cellular resolution |
Known theory says this cannot be solved by naive fitting:
- Degeneracy. Very different parameter sets produce the same activity (Prinz, Bucher & Marder 2004; Marder & Goaillard 2006).
- Sloppiness. In multi-parameter biophysical models, behavior depends on a few stiff parameter combinations and is insensitive to most others (Gutenkunst et al. 2007).
- Connectivity alone underdetermines dynamics (Beiran & Litwin-Kumar 2024). Recording a small subset of neurons can remove the degeneracy. The number of neurons needed scales with the dimensionality of the dynamics, not with network size.
- The anatomical connectome is incomplete as a signaling graph. In C. elegans, optogenetic signal propagation differs from what anatomy predicts, partly because of extrasynaptic neuropeptide signaling. The measured functional atlas predicts spontaneous dynamics better than anatomy-based models do (Randi et al. 2023). Drosophila EM connectomes do not report gap junctions.
Current state of the art (references.md) either:
- (a) uses simplified neurons and fits or tunes a few global parameters (Shiu et al. 2024 LIF brain; Lappalainen et al. 2024 task-optimized visual system; FlyGM 2026 graph controller), or
- (b) uses detailed biophysics at small scale with hand-tuned parameters (BAAIWorm / MetaWorm 2024).
Neither approach has a principled way to determine θ at scale.