Rectified Spectral Units (ReSUs): Neural Primitives

• ReSUs are biologically inspired neural primitives that use CCA-based projections to capture temporal and spatial dependencies in sensory data.
• They apply half-wave rectification to split signals into ON/OFF channels, emulating graded responses in post-photoreceptor circuits.
• Local Hebbian/anti-Hebbian learning enables multilayer ReSU networks to progressively extract complex features, mirroring Drosophila sensory processing.

Rectified Spectral Units (ReSUs) constitute a biologically inspired neural computational primitive designed to capture hierarchical temporal and spatial dependencies in sensory data by combining canonical correlation analysis (CCA) with rectifying nonlinearities. Each ReSU maintains a memory of recent inputs, projects this temporal window onto learned canonical directions that maximize predictive past–future correlation, and employs half-wave rectification to split responses into ON and OFF channels. This architecture enables self-supervised, local learning without error backpropagation, supporting the progressive extraction of increasingly complex features in multilayer networks and closely mirroring key functional properties of post-photoreceptor neuronal circuits in Drosophila (Qin et al., 29 Dec 2025).

1. Architectural Foundations of Rectified Spectral Units

A single Rectified Spectral Unit processes a scalar input time series $\{x_t\}$ by forming a fixed-length past window vector

$$\mathbf{p}_t = [x_t, x_{t-1}, \dots, x_{t-m+1}]^\top \in \mathbb{R}^m,$$

which is projected onto a learned canonical direction $w \in \mathbb{R}^m$: $u_t = w^\top \mathbf{p}_t.$ The canonical direction $w$ is determined to maximize the correlation between $\mathbf{p}_t$ and a future window $\mathbf{f}_t = [x_{t+1}, \dots, x_{t+h}]^\top$. This objective is formulated via rank-1 CCA, with covariances $\Sigma_{pp}$, $\Sigma_{ff}$, and $\Sigma_{fp}$ capturing, respectively, the statistics within past windows, future windows, and their cross-covariance. Whitening and singular value decomposition yield the optimal $w$ as

$w = \Sigma_{pp}^{-1/2} v_1,$

where $v_1$ is the leading right-singular vector of the normalized cross-covariance. For multi-channel ReSUs, projection is generalized with a linear map $\Psi = V_r^\top \Sigma_{pp}^{-1/2}$, providing $r$ canonical features. The truncated CCA maximizes the mutual information,

$$I_r = -\tfrac{1}{2} \sum_{i=1}^r \log(1-\sigma_i^2),$$

with $\sigma_1, \sigma_2, \dots$ the top canonical correlations.

2. Nonlinear Rectification Mechanism

Following linear projection, each channel is split into ON and OFF units using half-wave rectification: $$z^+_{t,i} = \max\{u_{t,i}, 0\}, \qquad z^-_{t,i} = \max\{-u_{t,i}, 0\}.$$ This models the physiological synaptic gating of continuous (graded-potential) neuronal outputs, where only depolarizations above or below baseline are transmitted downstream. Such rectification is analogous to the functional motif empirically observed in both insect and vertebrate sensory circuits, offering a biophysically plausible activation function distinct from standard artificial neural network nonlinearities.

3. Local Learning via Self-Supervised Hebbian/Anti-Hebbian Dynamics

ReSU learning operates strictly locally and self-supervised, leveraging moving-window estimates of the relevant covariances: $$\Sigma_{pp}(t) \leftarrow (1-\eta) \Sigma_{pp}(t-1) + \eta \mathbf{p}_t \mathbf{p}_t^\top,$$ and analogously for $\Sigma_{ff}$ and $\Sigma_{fp}$, with $\eta$ a forgetting factor. The update rule for $w$ uses a biologically inspired Hebbian/anti-Hebbian dynamic: $$\Delta w = \epsilon \left( \Sigma_{pp}^{-1} \Sigma_{pf} w - (w^\top \Sigma_{pf} w) w \right),$$ supplemented by normalization ($w^\top \Sigma_{pp}w=1$) to ensure unit variance in the projected direction. The first term aligns $w$ with the most predictive past–future structure, and the second orthogonalizes and normalizes the solution, paralleling competitive learning in unsupervised neural models.

4. Multilayer Construction and Drosophila Circuit Analogy

ReSUs can be composed into multilayer feedforward networks where each layer extracts higher-order features from its inputs:

• First Layer (Pixel-Driven ReSUs): Each unit receives temporal sequences from a single pixel (contrast scan over time), with parameter choices $m=75$, $h=50$. CCA yields a low-pass filter (first canonical direction; analog of L3) and, after rectification, ON (L1) and OFF (L2) band-pass derivative filters. Notably, these temporal kernels adapt to input signal-to-noise ratio (SNR), collapsing from multi-lobed (low noise) to single-lobed (high noise), as observed in Drosophila physiology.
• Second Layer (Spatial Pooling and Direction Selectivity): Inputs are concatenated ON/OFF signals from three proximal pixels:

$$\mathbf{y}_t = [z^+_{1,L}, z^-_{2,L}, z^+_{1,C}, z^-_{2,C}, z^+_{1,R}, z^-_{2,R}]^\top.$$

Performing CCA on $$(\mathbf{p}_t^2 = \mathbf{y}_t,\, \mathbf{f}_t^2 = \mathbf{y}_{t+\Delta})$$ (with $\Delta$ matching motion lag), the network discovers spatiotemporal filters whose weights and direction selectivity recapitulate properties of T4 cells (direction-selective neurons), including the sign and amplitude patterns of L1/L3 → T4a synapses established in connectomic circuits.

5. Experimental Characterization and Comparative Evaluation

Empirical evaluation addresses both computational and biological fidelity:

• Filter Shapes and Predictive Power: Leading canonical correlations are $\sigma_1 \approx 0.9$ (low-pass), $\sigma_2 \approx 0.6$ (band-pass). The mutual information $I_2$ indicates that two channels suffice to capture most predictive information between past and future windows.
• SNR Adaptation Dynamics: Increasing Gaussian noise in the inputs induces a rapid collapse of the second filter from multi- to single-lobed shape, matching shifts seen in Drosophila L1/L2 neurons. Adaptation to new optimal filters occurs within approximately 10 memory window lengths.
• Physiological Trace Comparisons: Outputs of the first- and second-layer ReSUs match empirical calcium imaging data for L3 (low-pass) and L1/L2 (rectified derivative) under staircase contrast stimulation, with $R^2 \gtrsim 0.8$.
• Direction Selectivity: The second-layer T4-analog ReSU achieves a Direction Selectivity Index (DSI) of $\approx 0.7$, consistent with measurements in biological T4 cells, with direction preference arising entirely from two-layer, self-supervised CCA learning—absent any backpropagation.

6. Theoretical and Computational Significance

Stacking CCA-based units with biologically plausible rectification yields a framework for constructing deep self-supervised networks with entirely local learning rules. This contrasts with conventional artificial neural networks employing backpropagation and ReLU activations. In Drosophila, ReSUs match empirical connectomic and physiological properties of L1–L3 and T4 neurons solely by maximizing predictive information through local statistics, supporting their utility as a modeling tool for sensory circuits and as a paradigm for biologically grounded machine learning (Qin et al., 29 Dec 2025).

A plausible implication is that ReSUs may generalize to more complex sensory domains and permit the construction of deep/local self-supervised models that maintain theoretical tractability and biological interpretability.