Hierarchical Neural Circuit Theory of Normalization and Inter-areal Communication

Abstract

The primate brain exhibits a hierarchical, modular architecture with conserved microcircuits executing canonical computations across reciprocally connected cortical areas. Though feedback connections are ubiquitous, their functions remain largely unknown. To investigate the role of feedback, we present a hierarchical neural circuit theory with feedback connections that dynamically implements divisive normalization across its hierarchy. In a two-stage instantiation (V1 ↔ V2), increasing feedback from V2 to V1 amplifies responses in both areas, more so in the higher cortical area, consistent with experiments. We analytically derive power spectra (V1) and coherence spectra (V1-V2), and validate them against experimental observations: peaks in both spectra shift to higher frequencies with increased stimulus contrast, and power decays as 1/f ⁴ at high frequencies (f). The theory further predicts distinctive spectral signatures of feedback and input gain modulation. Crucially, the theory offers a unified view of inter-areal communication, with emergent features commensurate with empirical observations of both communication subspaces and inter-areal coherence. It admits a low-dimensional communication subspace, where inter-areal communication is lower-dimensional than within-area communication, and frequency bands characterized by high inter-areal coherence. It further predicts that: i) increasing feedback strength enhances inter-areal communication and diminishes within-area communication, without altering the subspace dimensionality; ii) high-coherence frequencies are characterized by stronger communication (ability to estimate neural activity in one brain area from neural activity in another brain area) and reduced subspace dimensionality. Finally, a three-area (V1 ↔ V4 and V1 ↔ V5) instantiation of the theory demonstrates that differential feedback from higher to lower cortical areas dictates their dynamic functional connectivity. Altogether, our theory provides a robust and analytically tractable framework for generating experimentally-testable predictions about normalization, inter-areal communication, and functional connectivity.

Figure 1:. Schematic of the two-stage recurrent circuit model.

Solid and dashed lines represent excitatory and inhibitory connections, respectively. Principal excitatory neurons are denoted by y, modulatory excitatory neurons by u, and modulatory inhibitory neurons by a and q. Subscripts 1 and 2 designate neurons in visual areas V1 and V2, respectively. z₁ denotes the input drive from the LGN to V1, a weighted sum of the responses of LGN neurons. Parameters β₁ and γ₁ modulate the input and feedback gain to V1, respectively.

Figure 2:. Comparison of theoretical predictions (left column) and experimental observations (right column) in V1 and V2.

All theoretical predictions are generated using the same baseline parameters (Table 2). a, Theoretical mean firing rates as a function of stimulus contrast (V1: dashed line, V2: solid line). b, Experimental mean firing rates of V1 and V2 (data from [90] and [91], replotted for comparison). The shaded area with a solid border indicates the 25th to 75th percentile range for V1, and the one with the dashed border indicates the same for V2. c, Theoretical V1 power spectra at various stimulus contrast levels. Power spectra were normalized using the equation $\frac{(\text{power}-\text{baseline})}{(\text{power}+\text{baseline})}$, where baseline power is taken to be spontaneous activity at 0% contrast. The peak power shifts towards higher frequency with increasing contrast. The inset shows the $\frac{1}{f^4}$ power law decay at high frequency. d, Experimental power spectra from macaque V1 (data from [22], replotted for comparison). e, Theoretical coherence spectra between maximally firing neurons in V1 and V2. The peak coherence shifts towards higher frequency with increasing contrast. f, Experimental coherence spectra from macaque V1-V2 (data from [22], replotted for comparison). g, Theoretical communication subspaces, prediction performance as a function of dimensionality for inter-area (circles) and within-area (squares) communication. Simulation results indicate that the inter-area communication subspace is lower dimensional than the within-area communication. h, Experimental communication subspaces from macaque V1-V2 (data from [32], replotted for comparison). The plotted prediction performance in both panels g and h is an average across different subsets of the source and target neural populations, and the shaded areas represent the standard error of the mean (SEM).

Figure 3:. Theoretical predictions: Modulating feedback (left) and input gain (right).

a,b, Firing rates as a function of contrast. Increasing either feedback gain or input gain enhances neural responses, with feedback gain showing a greater impact in the higher cortical area V2. c,d, V1 power spectra: 3% contrast. An alpha peak is observed for low feedback gain, but the peak diminishes with increasing feedback gain. The alpha peak is absent with input gain changes. e,f, V1 power spectra: 50% contrast. A consistent gamma peak is observed, which shifts toward higher frequencies with increasing feedback gain (e) and input gain (f). g,h, Coherence spectra: 3% contrast. A broad peak in the beta band is observed at low feedback gain, which vanishes at high feedback gain. No such peak is observed with changes in input gain. i,j, Coherence spectra: 50% contrast. A beta peak is observed for low contrasts, which shifts toward higher (gamma) frequencies with increasing feedback gain. No beta peak is observed for changes in input gain, but the gamma peak shifts toward higher frequencies with increasing input gain. k,l, Communication subspaces. Increasing feedback gain enhances inter-areal (circles) communication while decreasing within-area (squares) communication. Conversely, increasing input gain decreases both inter- and within-area communication.

Figure 4:. Theoretical prediction: Frequency-dependence of communication.

a, V1-V2 coherence spectra for different contrasts; same as in Fig. 2e. b, Prediction performance versus frequency at different contrasts, averaged across different subsets of the source and target populations. At low stimulus contrast (≈ 10 %), a peak in communication efficacy is observed around 20 Hz. As stimulus contrast increases, the preferred frequency of communication shifts towards higher frequencies (40 Hz). The magnitude of communication reaches a maximum around 40% contrast, before declining at higher contrast levels. Under conditions of very low contrast (< 10%), communication is mostly concentrated at very low frequencies. These frequency-specific trends in communication parallel the V1-V2 coherence patterns shown in panel a. c, Dimensionality of the communication subspace versus frequency, averaged across different subsets of the source and target populations. A notable dip in dimensionality occurs at frequencies corresponding to peaks in communication (panel b) and coherence (panel a). The shaded areas in panels b and c represent the standard error of the mean (SEM).

Figure 5:. Theoretical prediction: Feedback-dependent modulation of functional connectivity.

a, Feedback from V5 → V1 is stronger than feedback from V4 → V1, and V1 preferentially enhances communication with V5. b, Feedback from V4 → V1 is stronger than feedback from V5 → V1, and V1 preferentially enhances communication with V4. The plotted prediction performance is an average across different subsets of the source and target populations, and the shaded areas represent the standard error of the mean (SEM). These results demonstrate that top-down feedback can dynamically route information flow between cortical areas, i.e., modulating functional connectivity.

Figure 6:. Hypothesized mapping of computational components in the model onto the dendritic compartments of a pyramidal cell.

The input drive is hypothesized to be the weighted sum of feedforward inputs arriving at the distal basal dendrites. This is modulated by an input gain, corresponding to inhibition at the proximal basal dendritic trunk. In the apical tuft, the feedback drive represents a weighted sum of feedback inputs from higher cortical areas. This signal is amplified by a feedback gain within the distal apical trunk. The recurrent drive corresponds to the sum of lateral inputs on the proximal apical dendrites. The combined recurrent and feedback signals are then modulated by a recurrent gain at the proximal apical trunk. The overall synaptic current is a combination of these gain-modulated drives. The figure is taken from [185].

Figure 7:. Connectivity matrices in the hierarchical recurrent circuit model.

a, The V1 encoding matrix (W_zx), each row represents the tuning curve of a V1 neuron. b, The recurrent connectivity matrix for V1 (W₁₁). The V2 recurrent connectivity matrix W₂₂ has a similar structure. c, The Feedback connectivity matrix from V2 to V1 (W₁₂). The corresponding feedforward matrix from V1 to V2, (W₂₁), is its transpose. Since W₂₁ is symmetric, the feedforward and feedback matrices are identical (W₁₂ = W₂₁).