Neural Encoding through Hierarchical Amplitude Modulation

Abstract Hierarchical encoding is a structural element of the Free Energy Principle and related information-centric accounts of brain function, but a concrete circuit-level mechanism for it remains elusive. Here we examine Hierarchical Amplitude Modulation (HAM). In this computationally grounded scheme, information is encoded in the envelope of a carrier and slower brain rhythms multiplicatively modulate the amplitude of faster rhythms, creating a cascade of nested oscillatory envelopes. This multiplicative architecture naturally produces intermodulation frequencies (of the form f c + Σ a i f i ). It predicts that oscillation frequency bands should be log-spaced ( r ≳ 2–3, depending on the number of modulation layers) to avoid spectral overlap, as in constant-Q filter banks, consistent with the observed logarithmic spacing of canonical brain rhythms (with ratios ∼2–3). HAM’s log-uniform distribution yields a 1 /f global spectral profile and 1 /f α sideband spectra in the broadband “aperiodic” component, with the slope determined by modulation depth and band ratio. We then demonstrate modulation and demodulation in the laminar neural mass model (LaNMM), where a fast excitatory-inhibitory oscillator circuit couples with a slower cortical oscillator (Janse-Rit). Through the network’s intrinsic nonlinearities and cross-frequency coupling, amplitude modulation and demodulation are implemented. These results provide a novel circuit-level mechanism for hierarchical predictive coding, linking theoretical principles to observed spectral features of brain activity. Highlights We introduce HAM 1 : a principle for the organization of hierarchical information processing in the brain. Information is encoded through amplitude modulation, where faster rhythms are multiplicatively modulated by slower processes. Information can thus live in signals or their envelopes (and envelopes of envelopes). The required intermodulation structure for band protection, f c ± Σ a i f i , finds a natural implementation in near geometric (log) spacing with a frequency ratio r ≳ 2–3 (strong super-increasing hierarchy), yielding constant fractional bandwidths and enabling staged demodulation. In addition to geometric frequency spacing, HAM provides two routes to 1 /f α : (i) a cascade link α = 2 ln(2 /m ) / ln r ; (ii) a log-uniform mixture that yields 1 /f in expectation with constant-Q kernels. We provide a proof of concept modulation and demodulation implementation in NMM using the LaNMM with PING-like fast generators as carriers and JR-like slow generators for envelope extraction.