NeuroTwin NFC Mathematical Constitution#

This is the canonical markdown entry point for the NeuroTwin NFC math. The longer LaTeX/PDF dossier remains in docs/research/neurotwin_nfc_research_dossier.tex and docs/research/neurotwin_nfc_research_dossier.pdf.

Primitive#

NFC treats each recording as a partial, noisy observation of a subject-specific latent neural field:

\[F_s(x,t,\omega) \in \mathbb{R}^d\]

Here s indexes subject, x indexes neural location or parcel, t indexes time, and omega captures stochastic state.

Observation Operator#

Each modality is an observation operator over the same field:

\[Y_m = \mathcal{O}_m(F_s,A_s,U,\epsilon_m)\]

A_s is anatomy or subject geometry, U is stimulus/task/context, and epsilon_m is modality-specific noise.

Generative Model#

\[p(Y_{1:M}\mid U,A_s)=\int p(F\mid U,A_s)\prod_m p(Y_m\mid F,A_s)\,dF\]

The benchmark question is whether a field-mediated model explains held-out observations better than direct translation baselines.

Controlled Dynamics#

\[F(t+\Delta)=\Phi_{\theta,\Delta}(F(t),U_{[t-H,t]},A_s)\]

This says future field state depends on current field state, recent stimulus/context history, and subject anatomy.

Neural Field Dynamics#

\[\tau \partial_t F(x,t) = -F(x,t) + \int_\Omega K_\theta(x,x',t)\sigma(F(x',t))\,dx' + B_\theta U(t) + \eta(x,t)\]

Discretized Dynamics#

\[Z_{t+1}=Z_t+\Delta t[-DZ_t+K_t\sigma(Z_t)+BU_t]+\xi_t\]

Low-Rank Pair Kernel#

\[K_t \approx U_tV_t^\top\]
\[M_t=\operatorname{softmax}\left((U_tV_t^\top)/\sqrt r+S\right)\]
\[Z'_t=Z_t+M_tZ_tW\]

This is where the old Pair-Operator idea survives: not as the main architecture, but as a low-rank relational field-update ablation.

fMRI Observation#

\[(H_{\mathrm{HRF}}a)(t)=\int_0^\infty h(\tau)a(t-\tau)\,d\tau\]
\[Y_{\mathrm{fMRI}}(p,t)=R_pH_{\mathrm{HRF}}g_\theta(F(\cdot,t))+\epsilon\]

EEG and MEG Observation#

\[Y_{\mathrm{EEG}}(t)=L_sJ_\theta(F_t)+\epsilon_{\mathrm{EEG}}\]
\[Y_{\mathrm{MEG}}(t)=M_sJ_\theta(F_t)+\epsilon_{\mathrm{MEG}}\]

Spike, Calcium, and Behavior Observations#

\[Y_{n,t}\sim\operatorname{Poisson}\left(\Delta t\cdot\operatorname{softplus}(w_n^\top F(x_n,t))\right)\]
\[c_{n,t}=(k_{\mathrm{Ca}}*r_n)(t)+\epsilon\]
\[p(a_t\mid F_t,U_t)=\operatorname{softmax}(C\operatorname{pool}(F_t)+DU_t)\]

These are theory entries unless corresponding adapters and tests are explicitly implemented.

Operator Learning Interpretation#

NFC is not direct modality fusion. It learns an inverse path from observations to latent field state and a forward path from field state to modality-specific readouts.

State-Space Interpretation#

NFC can be read as a controlled state-space model where F_t is the latent state and each modality supplies a partial observation channel.

Graph Calculus Interpretation#

\[\nabla_wF(i,j)=\sqrt{w_{ij}}(F_j-F_i)\]
\[L=D-W\]
\[R_{\mathrm{graph}}(F)=\sum_{(i,j)\in E}w_{ij}\|F_i-F_j\|^2=\operatorname{Tr}(F^\top L F)\]

Identifiability and Gauge Ambiguity#

\[F'=AF\]
\[\mathcal{O}'_m=\mathcal{O}_mA^{-1}\]
\[\mathcal{O}'_m(F')=\mathcal{O}_m(F)\]

Latent fields are identifiable only up to transformations unless constrained by architecture, observations, and regularizers.

Uncertainty and Calibration#

\[\mathcal{L}_{\mathrm{NLL}} = \sum_{m,t,i} \frac{(y_{m,t,i}-\mu_{m,t,i})^2}{2\sigma_{m,t,i}^2} + \frac12\log\sigma_{m,t,i}^2\]
\[\mathbb{P}[Y\in C_\alpha(X)]\approx 1-\alpha\]

Uncertainty artifacts are not claim evidence unless they use actual uncertainty outputs and a documented calibration target.

Synthetic Proving Ground#

The NFC synthetic suite is a gate, not a result. It must test true field-grounded tasks, no-observation and no-pair ablations, no NaNs, strict shape contracts, and no target leakage.

Why Direct Fusion Is Not Enough#

Direct fusion can predict one modality from another, but it does not force a shared field explanation. NFC’s scientific bet is that field-mediated translation is more robust under held-out subject/site/dataset splits.

Why Pair-Operator Is a Submodule#

Pair-Operator captures low-rank relational updates. NFC needs that idea only as one possible kernel inside a broader latent-field and observation-operator model.