Neuron Interaction and Nowcasting (NiNo)

• NiNo is a domain that integrates statistical, algorithmic, and modeling techniques to infer and nowcast dynamic neural interactions from ensemble spiking data.
• It combines Bayesian inference, dynamical models, nonlinear point processes, and graph neural networks to accurately reconstruct connectivity and predict near-term activity.
• Validated on synthetic and real neural datasets, NiNo methods demonstrate robust performance, scalability, and biological plausibility in forecasting brain dynamics.

Neuron Interaction and Nowcasting (NiNo) encompasses a set of statistical, algorithmic, and modeling approaches for inferring, forecasting, and interpreting dynamic interactions among neurons from ensemble spiking data. The field integrates Bayesian inference, dynamical systems models (including stochastic integrate-and-fire and nonlinear point processes), network theory, and deep learning methods adapted for neural population analysis. NiNo frameworks systematically leverage both the explicit structure of neuronal circuits and latent processes such as noise, variable memory, and stimulus-driven modulations, thus providing rigorous mechanisms for both reconstructing connectivity and predicting the system's near-term activity state.

1. Bayesian Inference of Neuronal Couplings

Bayesian strategies for NiNo, as exemplified in models of assemblies of stochastic integrate-and-fire neurons (Monasson et al., 2011), focus on simultaneous recovery of synaptic interactions and external currents from spike trains. The leaky integrate-and-fire (LIF) model governs the subthreshold membrane dynamics,

$$C \frac{dV_i}{dt} = -g V_i + I_i^\mathrm{syn}(t) + I_i + \eta_i(t)$$

with instantaneous synaptic integration (a Dirac delta spike in $I_i^\mathrm{syn}(t)$) and Gaussian noise term $\eta_i(t)$. Two Bayesian procedures are used:

• Fixed Threshold procedure: Assumes a constant firing threshold, computes the likelihood via optimal deterministic membrane trajectories, and identifies "active" or "passive" contact points for exact likelihood maximization in the vanishing-noise regime.
• Moving Threshold procedure: Heuristically adjusts the spike threshold as a function of trajectory survival probability for moderate noise, yielding a time– and context–dependent threshold, $$V_{\mathrm{th}}^M = V_{\mathrm{th}} + [2(dp_s/dV)_{V=V_{\mathrm{th}}}]^{-1}$$, which better models the influence of noise.

The Bayesian log-likelihood is quadratic in the parameters,

$$L^* = -\frac{1}{2}\sum_{i,k}\int_{t_{ik}}^{t_{i(k+1)}}dt[\eta_i^*(t)]^2,$$

and parameter inference proceeds via Newton–Raphson optimization, exploiting factorization over interspike intervals (ISIs). Error estimates from the Hessian of $L^*$ scale as $S^{-1/2}$, with $S$ the spike count, providing quantitative confidence in the inferred couplings.

2. Model Validation and Influence of Biophysical Dynamics

Validation on both synthetic networks (with ground-truth couplings) and experimental retinal ensemble recordings demonstrates that both procedures recover directional connectivity, with inference accuracy controlled by spike count and membrane time constants ($\tau = C/g$). Notably:

• Large $\tau$: Accurate inference; potential evolves nearly linearly.
• Small $\tau$: Exponential decay dominates; inference of negative interactions becomes sensitive to latency, governed by $$J_{ij} \sim \text{const} \cdot \exp(\text{latency}/\tau)$$.
• The amplitude and character of inferred couplings reflect the biophysical regime of passive versus active contacts and the temporal scale of synaptic effects.

The Bayesian approach contrasts with classical cross-correlation methods, which yield symmetric (non-causal) correlation matrices and cannot resolve directed or mediated (third-neuron) effects.

3. Variable Memory Models and Consistent Model Selection

NiNo frameworks have expanded to stochastic point-process models with variable-length memory, where each neuron's spiking probability is determined by a combination of its post-spike reset and contributions from presynaptic neurons, weighted by the synaptic matrix $W = \{\omega_{j\to i}\}$ (Ferreira et al., 12 Nov 2024).

• MLE for Synaptic Weights: Each neuron is optimized via a separable logistic regression,

$$\ell(W;\mathbf{x}) = \frac{1}{T}\sum_{i=1}^N\sum_{t=1}^T [x_t(i)v_{t-1}(i) - \log(1+\exp(v_{t-1}(i)))]$$

with consistency proofs ensuring convergence of the estimator to true parameters as $T\to\infty$.

• Neighborhood Selection: Sensitivity measures quantify the impact of omitting each presynaptic neuron $j$; the neighborhood inference is consistent in both false positive and false negative rates, rooted in the regularity of transition probabilities and ergodic theorems.

Simulation studies and analysis of hippocampal visual task data demonstrate the recovery of biologically plausible connectivity graphs and dynamic spike patterns.

4. Flexible Interaction Modeling via Nonlinear Point Processes

Classic Hawkes processes, constrained to excitation, are generalized via sigmoid nonlinear Hawkes models (Zhou et al., 2020), incorporating both excitatory and inhibitory interactions by modeling the conditional intensity as

$$\lambda_i(t) = \bar{\lambda}_i \cdot \sigma(h_i(t)),$$

with $h_i(t)$ composed of weighted sums of pre-synaptic spike convolved with nonnegative basis functions. Posterior analytic tractability is achieved by Polya–Gamma augmentation, latent Poisson marks, and sparsity-inducing measures, facilitating closed-form EM updates for the entire interaction weight vector $w_i$.

The methodology yields rapid recovery of sparse functional connectivity matrices from large-scale cortical spike trains and is fundamentally suited for nowcasting via near-term forecasting of spiking activity.

5. Disentangling Nonstationary Inputs and True Interactions

Discriminating the origin of correlations (common input vs. true coupling) is a critical problem (Tyrcha et al., 2012). Kinetic Ising models with explicit time-dependent fields $h_i(t)$ and couplings $J_{ij}$ enable machine learning approaches (gradient-based likelihood maximization) to compare model fits under various assumptions:

• Retinal data: Most correlations stem from external stimulus nonstationarity; couplings inferred in nonstationary models are weak and unstable.
• Cortical simulation: Inclusion of $J_{ij}$ in a nonstationary field model captures genuine connectivity; model selection metrics (AIC, noise–signal ratios) and Zipf's law for synchronous patterns provide systematic quantification of model quality.

Explicit modeling of nonstationarity is essential for accurate nowcasting in real data, preventing misattributed connectivity.

6. Network-Theoretic and Dynamical Perspectives on Emergent Patterns

NiNo approaches also leverage kinetic Ising-inspired lattice models and network theory to probe the emergence of functional cortical patterns (FCPs) at phase transitions (Gund et al., 2021):

• Coupling strength $J$ and interaction range $n$ (via $J(r_{ij}, n) = J/r_{ij}^n$) together induce scale-free, hierarchically organized domains of activity upon entering the critical regime.
• Topological measures (clustering coefficients, degree distributions, centralities, rich-club analysis) and multifractal detrended fluctuation analysis yield quantitative congruence with EEG-derived functional networks.
• The approach permits nowcasting of cognitive states by tracking critical temperature $T_c$ and interaction range parameters.

7. Deep Learning and Graph Networks for Neural Training Acceleration

Recent advances apply NiNo principles in deep learning, notably for accelerating neural network training via graph-based parameter nowcasting (Knyazev et al., 6 Sep 2024). Here, neuron connectivity is encoded via neural graphs, processed by graph neural networks (GNNs), and used to forecast weight trajectories beyond conventional optimizers (Adam). Crucial architectural adaptations address permutation symmetries in Transformer models and distinguish roles of various weight matrices (e.g., $W^q$, $W^k$, $W^v$, $W^o$).

Empirical results on vision and language tasks indicate up to 50% reduction in training steps versus standard optimizers, with robust performance across network types.

8. Biological Realism and Functional Connectivity Discovery

Biologically inspired architectures, such as SynapsNet (Delavari et al., 12 Nov 2024), model population activity by combining time-invariant neuron embeddings, learnable directed functional connectivity matrices $A$, and shared dynamical decoders (GRUs). Input current for each neuron is linearly weighted by connectivity,

$I_t^{(i)} = \sum_k A^{i,k} X_t^{(k)},$

with subsequent population activity forecasts

$$\hat{X}_{t+1}^{(i)} = f(X_{t-W+1:t}^{(i)}, I_{t-W+1:t}^{(i)}, B_{t-W+1:t}, E^{(i)}).$$

Benchmarks on calcium imaging and Neuropixels datasets demonstrate consistent outperformance over sequential RNNs/LSTMs and other deep models, and the inferred connectivity matrix is both sparse and directional, facilitating interpretable predictive interactions. Cross-correlation and synthetic ground-truth experiments confirm robust discovery of functional connections.

Summary

NiNo encompasses a suite of mathematical, algorithmic, and biological frameworks for the inference, selection, forecasting, and interpretation of neural interactions. Approaches span Bayesian inversion of integrate-and-fire models, nonlinear Hawkes processes with analytic EM inference, machine learning for model disentanglement of correlation sources, network-theoretic characterization of functional cortical patterns near phase transitions, and graph-driven deep learning for both neural training acceleration and population dynamics modeling. Recent methods achieve high accuracy in predicting both network activity and directed connectivity, with rigorous error controls, statistical consistency proofs, and quantitative validation on both synthetic and large-scale real neural datasets. The field's continued methodological evolution will further unify biological realism, computational efficiency, and interpretability in nowcasting complex brain dynamics.