NeuroFisherSNR

This repository contains the implementation for the paper:

Quantifying Signal-to-Noise Ratio in Neural Latent Trajectories via Fisher Information. European Signal Processing Conference. EUSIPCO, Lyon, France. Hyungju Jeon, Il Memming Park arXiv:2408.08752

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Overview

Neural spiking activity is inherently stochastic, with significant variability across different trials and neurons. Yet, buried within this noise lies critical information about population-wide latent states—the hidden neural dynamics that drive and are driven by behavior and perception. To assess the feasibility of accurately extracting these latent states from a given dataset, we need statistical tools that help us understand how much information neural spikes carry.

Given neural spiking recordings, how much do the neurons inform us about the latent neural states?

To answer this question, we use Fisher Information to derive an upper bound for the Signal-to-Noise Ratio (SNR) of the latent trajectories observed from spikes.

We model neural spiking activity of neuron $i$ at time $t$, $y_i(t) \in \mathbb{N}_0$, with a log-linear Poisson state space model:

$$y_i(t) \sim Poisson(\lambda_i(t))$$

where the firing rate $\lambda_i(t) > 0$ is defined as the exponential of a linear function of the latent trajectory $\mathbf{x}(t) \in \mathbb{R}^d$ and the observation model parameters (loading vector $C_i \in \mathbb{R}^{1 \times d}$ and bias $b_i \in \mathbb{R}$):

$$\lambda_i(t) = \exp(C_i \mathbf{x}(t) + b_i)$$

The Fisher Information of the latent trajectory given a population of $n$ neurons is:

$$\mathbf{I}^{{\mathrm{pop}}}(\mathbf{x})=\mathbf{C}^{\top}\text{diag}(\boldsymbol{\lambda}(\mathbf{x}))\mathbf{C}$$

where $\boldsymbol{\lambda}(\mathbf{x}) = (\lambda_1(\mathbf{x}), \lambda_2(\mathbf{x}), \ldots, \lambda_n(\mathbf{x}))^{\top}$ is the vector of firing rates for all neurons, and $\mathbf{C} = [C_1, C_2, \ldots, C_n]^{\top} \in \mathbb{R}^{n \times d}$ is the loading matrix.

We can then derive the SNR of the latent trajectory using the Cramér-Rao lower bound:

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Core Features

• Generate Poisson spike train observations and log-linear observation model parameters with controllable signal-to-noise ratio (SNR)
• Compute Instantaneous Observed Fisher Information
• (To be implemented) Estimate Cramér-Rao lower bounds on estimation accuracy for the latent trajectory
• (To be implemented) Evaluate and compare inference results with theoretical uncertainty

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Installation

Clone and run from source:

git clone https://github.com/hyungju-jeon/neuro-fisher.git
cd neuro-fisher

pip install -r requirements.txt

or

conda env create -f environment.yml

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Usage

Generate Poisson spike train observations and log-linear observation model parameters with a target signal-to-noise ratio (SNR) given a latent trajectory.

observations, C, bias, firing_rate_per_bin, snr = gen_poisson_observations(
    x=latent_trajectory,
    C=None,
    d_neurons=num_neurons,
    tgt_rate_per_bin=target_rate_per_bin,
    max_rate_per_bin=max_rate_per_bin,
    priority="max",
    p_coh=p_coherence,
    p_sparse=p_sparse,
    tgt_snr=target_snr,
)

• x if the latent trajectory of interest in shape (n_timepoints, d_latent). It should be normalized to have unit variance and zero mean. If not, it will be normalized to have unit variance and zero mean.
• C is the loading matrix in shape (d_neurons, d_latent). If C is provided, it will be used as the loading matrix but scaled to match the target signal-to-noise ratio (SNR) per bin. If C is not provided, it will be generated with target coherence and sparsity.
• d_neurons is the number of neurons.
• tgt_rate_per_bin is the target mean firing rate per bin.
• max_rate_per_bin is the maximum firing rate per bin.
• priority can be "mean" or "max". "mean" prioritizes optimizing the mean firing rate per bin while "max" prioritizes the maximum firing rate per bin.
• p_coh and p_sparse are the target coherence and sparsity of the loading matrix.
• tgt_snr is the target signal-to-noise ratio (SNR) per bin. Suggested value of SNR bound is around 10 * log10(tgt_rate_per_bin * d_neurons / d_latent * 2 * log(max_rate_per_bin / tgt_rate_per_bin))

Demo Scripts

Generate log-linear Poisson observations from a Gaussian Process (GP) latent trajectory with Fisher Information SNR bound:

python demo/simulate_observation_gp.py

Generate observations from a 2D oscillation latent trajectory:

python demo/simulate_observation_oscillation.py

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Citation

If you use this code in your research, please cite:

@InProceedings{Jeon2024b,
  author    = {Hyungju Jeon and Il Memming Park},
  booktitle = {European Signal Processing Conference},
  title     = {Quantifying signal-to-noise ratio in neural latent trajectories via {F}isher information},
  month     =  aug,
  year      = {2024},
  archivePrefix = "arXiv",
  primaryClass  = "q-bio.NC",
  eprint        = "2408.08752"
}