This short article is inspired by my experience reading computational neuroscience papers. A common strategy for making neural models more interpretable involves projecting high-dimensional neural activity into a lower-dimensional space. This makes intuitive sense. It is difficult to grasp the relationships between hundreds or thousands of neurons in their original high-dimensional setting.
A closer look at dimensionality reduction methods like PCA, ISOMAP, and LDA reveals that these latent spaces are not entirely abstract. They are defined by explicit bases, which we can access and analyze. In doing so, we can begin to ask which neurons are responsible for shaping these low-dimensional dynamics.
Background
We begin with a data matrix: \(X \in \mathbb{R}^{N \times T}\) where $N$ is the number of neurons and $T$ is the number of timepoints.
PCA proceeds by computing the singular value decomposition (SVD) of the mean-centered matrix $X$: \(\tilde{X} = U S V^\top\) where $U \in \mathbb{R}^{N \times N}$ contains the left singular vectors (neuron-space directions), $S$ is a diagonal matrix of singular values (scaling each mode), and $V^\top \in \mathbb{R}^{T \times T}$ contains the right singular vectors (time-space directions).
The columns of $U$ define weighted combinations of neurons that form axes in the latent space. The columns of $V$ describe how each component evolves over time, and the singular values in $S$ represent the amount of variance explained by each component.
Each principal component can be written as: \(PC_k(t) = \sigma_k \cdot u_k v_k^\top(t)\) This expresses the rank-1 spatiotemporal pattern captured by the $k^\text{th}$ component. The vector $u_k$ lies in neuron space, indicating the contribution weights of each neuron. The vector $v_k^\top(t)$ describes the temporal evolution of this pattern, and $\sigma_k$ reflects how important the component is in terms of variance.
The first principal component represents the dominant joint activity pattern, which is a specific combination of neurons that varies together over time in a meaningful way. So while PCA summarizes population activity in low dimensions, it does not explicitly tell us which neurons contribute when. The entries $U_{ik}$ tell us how much neuron $i$ contributes to the $k^\text{th}$ principal axis, but this contribution is fixed and does not evolve with time.
Constructing a Time-Resolved Contribution Basis
Here I propose a simple but useful idea: we can decompose the expression of a latent trajectory over time to trace how individual neurons contribute to it. Once high-dimensional neural activity is projected into a lower-dimensional space using PCA, we don’t need to treat the latent dimensions as abstract summary variables. Each principal component is a linear combination of neurons and we can quantify how much each neuron contributes to its expression at each moment in time.
Suppose we have $N$ neurons and have applied PCA to reduce the activity to $K$ dimensions. Let $x_i(t)$ be the activity of neuron $i$ at time $t$, and let $u_{ik}$ be the loading of neuron $i$ on the $k^\text{th}$ principal component. Then the projection of the full population activity onto PC $k$ is given by:
\[z_k(t) = \sum_{i=1}^N u_{ik} \cdot x_i(t)\]This is a time-varying latent signal, often interpreted as a summary of population dynamics.
To trace where this latent signal comes from, we define the contribution of neuron $i$ to PC $k$ at time $t$ as:
\[C_{ik}(t) = u_{ik} \cdot x_i(t)\]This gives a time-resolved, neuron-specific breakdown of the latent trajectory. At each timepoint, the sum of contributions across all neurons gives back the full projection:
\[z_k(t) = \sum_i C_{ik}(t)\]In this way, we convert each latent dimension into a temporally evolving pattern of contributions distributed across neurons.
Figure: PCA decomposes population activity into latent trajectories. The contribution basis breaks these trajectories down into neuron-specific roles over time.
If desired, we can normalize each $C_{ik}(t)$ by the total projection to get a relative contribution:
\[\text{RelativeContribution}_{ik}(t) = \frac{C_{ik}(t)}{z_k(t)}\]This expresses the proportion of the latent expression at that moment attributable to a specific neuron. Because principal components can have both positive and negative loadings, the sign of the contribution carries important information and should not be ignored.
Though this idea is simple, I haven’t seen it formalized in the literature. Most PCA-based analyses treat latent variables as static descriptors or behavioral correlates, without asking how the expression of these components arises from the original neural activity. By tracing the time-resolved contributions of individual neurons, we gain a much clearer view of how roles evolve during behavior, and how neural circuits reshape their collective dynamics in real time.
Why This Matters for Dynamical Systems
This idea of tracing neuron contributions over time becomes especially important when thinking about neural activity as a dynamical system. In that framing, we are not just looking at neural activity as a static cloud of points. We are watching how population activity moves through state space. We care about trajectories, flows, and the directions the system bends into over time.
At one point, I tried to dive into this directly using real neural data from the Allen Brain Observatory. I was interested in how signals might propagate through a population, especially during natural movie stimuli. The idea was to iteratively estimate the Jacobian, the local linearization of the system, and use it to infer how activity at time $t$ influenced activity at time $t+1$. But I quickly ran into a wall. The Jacobians were huge, noisy, and nearly impossible to interpret. Even when I applied dimensionality reduction, I could not figure out how to relate the compressed trajectories back to specific neurons. I ended up with abstract dynamics that I could not pin to anything biologically meaningful.
That failure made me step back and rethink the entire pipeline. I realized I was trying to read the system’s behavior before I had identified its parts. That led me to this much simpler idea: if we start with the latent space, can we at least say who is responsible for what? Can we trace a latent trajectory back to the neurons driving it, and see how those roles change as time unfolds?
Toward Understanding Attractors in Neural Systems
One possible application of this approach is in studying attractor dynamics, the idea that neural populations may converge toward stable activity patterns over time, depending on task, stimulus, or internal state. Attractors are usually studied in latent space or inferred from global population patterns, but what if we could instead track them through the evolving roles of individual neurons?
If we compute the contribution matrix $C(t)$ , where each entry is:
\[C_{ik}(t) = u_{ik} \cdot x_i(t)\]then we can define a time-varying contribution vector for each neuron:
\[\vec{c}_i(t) = [C_{i1}(t), C_{i2}(t), \dots, C_{iK}(t)]\]This gives us a $K$-dimensional trajectory that describes how neuron $i$’s influence on the latent space evolves over time. Doing this for all neurons produces a structured view of how the population reorganizes itself to drive state transitions.
From here, we can define a pairwise influence function between neurons:
\[A_{ij}(t) = \text{sim}(\vec{c}_i(t - \tau), \vec{c}_j(t))\]where $\text{sim}(\cdot)$ could be a correlation, dot product, or another causality metric, and $\tau$ is a time lag. This function can be visualized as a dynamic influence heatmap, where the activation of one neuron in latent space appears to modulate the future expression of another.
By examining these influence maps over time, it may be possible to detect repeating patterns, convergent configurations, or cyclical loops. These are all hallmarks of attractor dynamics. Unlike traditional phase-space analyses, this framework preserves neuron identity, which means we can now ask: which neurons initiate transitions into attractor basins? Which ones stabilize them? And how does this vary across trials or behavioral conditions?
This kind of neuron-level tracing could offer a new lens on what it means for a brain region to stabilize, shift, or reverberate.