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Scalable inference of functional neural connectivity at submillisecond timescales
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Proposes continuous-time Poisson point-process GLM fitting via Monte Carlo CIF estimation and a quadratic polynomial approximation, with generalized Laguerre temporal bases, enabling submillisecond functional connectivity inference on large hippocampal recordings.
Problem
Standard Poisson GLMs for spike trains bin spike times, limiting temporal resolution to ~1 ms and making design matrices too large for large-scale recordings; the continuous-time likelihood's cumulative intensity integral is intractable.
Approach
Approximate the continuous-time CIF integral either by stratified Monte Carlo sampling (MC) or by a second-order polynomial approximation (PA) yielding a closed-form quadratic objective, and use exponentially scaled generalized Laguerre polynomials as orthogonal temporal basis with closed-form integrals; a hybrid PA-initialized MC warm start is also proposed.
Key results
- On simulated all-to-one data, MC and hybrid PA-MC models achieve the best filter recovery, with PA-c and PA-d less accurate, and continuous methods' runtime largely insensitive to recording length.
- Gauss-Lobatto quadrature baseline on N=8, T=100 s data reached MSE 5.37±0.48 vs MC 2.54±0.75, with quadrature taking ~2.5 hours vs ~20 seconds for MC.
- On hippocampal data, hybrid PA-MC recovers filters matching CCGs with only 500 s of data, and CA3 recurrent excitatory connectivity is ~3.83% of pairs, consistent with anatomy.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use MC sampling instead of Gauss-Lobatto quadrature | Continuous-time Poisson GLM · Simulated GLM, N=8, T=100 seconds | filter MSE: 5.37± 0.48 -> 2.54± 0.75 | Helped |
| Use MC sampling instead of Gauss-Lobatto quadrature | Continuous-time Poisson GLM · Simulated GLM, N=8, T=100 seconds | runtime: approximately 2.5 hours -> ~20 seconds | Helped |
| Use hybrid PA initialization followed by MC finetuning instead of MC from scratch | Continuous-time GLM · Simulated data, varying recording length and population size | filter recovery accuracy | Helped |
| Use continuous-time PA instead of discrete-time PA | Polynomial-approximate GLM · Simulated data, varying recording length and population size | filter recovery accuracy | Helped |
| Use generalized Laguerre bases instead of raised cosine bases | Continuous-time MC GLM · Simulated all-to-one GLM | filter reconstruction error | Helped |
| Use generalized Laguerre bases instead of raised cosine bases | GLM · Single-probe hippocampal recording | held-out log-likelihood | Helped |
| Vary the generalized Laguerre basis hyperparameters c and α (ablation) | GLM · Simulated data | filter MSE | No clear effect |
| Use polynomial approximation instead of MC estimation | Continuous-time GLM · Simulated data | filter recovery accuracy | Hurt |
| Use hybrid PA-MC instead of PA-c or PA-d | Continuous-time GLM · Single-probe hippocampal recording | filter alignment with CCGs | Helped |
| Use discrete batched GLM instead of continuous-time methods | Poisson GLM · Simulated data, increasing population size | filter recovery accuracy | Hurt |
| Widen the polynomial approximation interval (ablation) | Polynomial-approximate GLM · Simulated and real spike-time data | approximation error | Hurt |
Surprising findings
- Discrete batched SVRG GLM still underperforms even when its runtime exceeds other models by orders of magnitude.
- PA-c slightly outperforms PA-d on simulated filters despite identical nonlinearity and approximation range, attributed to binning inaccuracies.
Limitations
- Polynomial approximation is a global approximation with approximation error; wider intervals increase error
- Connectivity inferred from GLM coupling filters cannot fully separate monosynaptic connections from common input; CCGs are only a proxy
- Poisson likelihood may be suboptimal for neural spike count variance
Experimental setup
Datasets: Allen Institute Visual Coding Neuropixels session 715093703 (hippocampal CA1, CA3, DG) simulated all-to-one and all-to-all GLM spike trains
Benchmarks: Allen Institute Visual Coding Neuropixels hippocampal data (CCG alignment)
Metrics: mean squared error of filters time to completion normalized gradient MSE cross-validated log-likelihood
Baselines: discrete batched GLM (SVRG) discrete polynomial approximation (PA-d) raised cosine basis GLM Gauss-Lobatto quadrature point-process GLM
Builds on: Poisson GLM polynomial approximate GLM (Chebyshev) SVRG Gauss-Lobatto quadrature point process GLM
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