CARFAC: Asymmetric Resonators with Fast-Acting Compression

• CARFAC is a neuromorphic auditory model that mimics human cochlear mechanics using cascaded biquadratic filters and fast-acting compression.
• It employs the Greenwood function for place-frequency mapping and leverages FPGA-based implementations for high-resolution, low-latency audio analysis.
• The design integrates dynamic gain control and neuromorphic feature extraction, facilitating robust machine hearing and efficient signal processing.

The Cascade of Asymmetric Resonators with Fast-Acting Compression (CARFAC) model is a neuromorphic auditory front-end designed to emulate the key mechanical and nonlinear properties of the human cochlea. It is built on a cascade of biquadratic pole–zero filters, which mimic the traveling wave and frequency selectivity of the basilar membrane (BM), and is augmented with fast-acting compression (FAC) stages inspired by outer hair cell (OHC) active gain control. CARFAC has been implemented in multiple hardware and software environments, including FPGAs for real-time, high-resolution applications in machine hearing, speech recognition, and underwater acoustics.

1. Mathematical Foundation and Filter Architecture

The CARFAC model structurally consists of a cascade of two-pole–two-zero (biquadratic) digital filters, each representing a localized mechanical segment of the BM. The resonance of each filter section is governed by the place-frequency mapping dictated by the Greenwood function:

$f(x) = 165.4 \left(10^{2.1x} - 1\right)$

Here, $x$ is the normalized cochlear position from apex (0) to base (1), providing the tonotopic frequency allocation central to cochlear mechanics.

The standard transfer function for a single CARFAC filter section is:

$$\frac{Y}{X} = g \cdot \frac{z^2 + (-2a_0 + h c_0) r z + r^2}{z^2 - 2a_0 r z + r^2}$$

where:

• $z$ is the z-transform variable,
• $a_0 = \cos(\mathrm{OR})$, $c_0 = \sin(\mathrm{OR})$ (with OR as the pole ringing frequency),
• $h$ tunes the zero position relative to the pole,
• $r$ dynamically sets pole and zero radii (damping),
• $g$ normalizes DC gain.

The coefficient constraints ensure physically meaningful, complex zeros especially for high-frequency channels:

$h < \frac{2 + 2a_0}{c_0}$

Dynamic gain control and nonlinearity (FAC) are introduced through local feedback modulation of parameters such as $r$, enabling compression akin to OHC-mediated feedback observed in biological cochleae. In FPGA implementations, special attention is paid to efficiently realizing these computations, often approximating ratios like the DC gain $g$ using quadratic polynomials for hardware efficiency (Thakur et al., 2015, Bremer et al., 11 Aug 2025).

2. Fast-Acting Compression and Nonlinear Feedback

Sensory adaptation and dynamic range compression in CARFAC are driven by cascaded gain control loops interfacing filter outputs with models of OHC and IHC activity. The OHC stage modulates the filter undamping factor $r$ and thereby the local frequency tuning and gain of each channel through automatic gain control (AGC), enabling large dynamic range and context-dependent selectivity.

For example, in hardware implementations for embedded systems, OHC and IHC nonlinearity functions originally written as divisions are replaced by polynomial or fixed-power approximations:

• OHC compression function may be implemented as:

$\mathrm{NLF} = [\max(0, 1 - \mathrm{sqr}/8)]^8$

• IHC nonlinearity approximation:

$v_{\mathrm{mem}} = 0.75 (1-p)^2$

where $p$ is the rectified and saturated BM velocity (Bremer et al., 11 Aug 2025).

Such approximations are critical for energy-efficient, low-latency FPGA design as they minimize resource usage and avoid expensive division operations.

3. FPGA Implementations and Algorithmic Optimizations

CARFAC has been realized on several FPGA platforms, including Xilinx Virtex-6 and AMD Kria KV260 SoM, each exploiting the inherent parallelism, pipelining, and time-multiplexing characteristics of hardware for high-resolution sound analysis.

• Virtex-6 implementation supports up to 1224 filter sections at 142 MHz clock frequency, completing all processing in less than 250 μs latency with input sound rates of 48 kHz (Thakur et al., 2015).
• On Kria KV260, time-multiplexing and pipeline designs enable concurrent processing of multiple channels (>64) with overall board power consumption measured at 3.11 W for real-time underwater acoustics at 256 kHz sample rates (Bremer et al., 11 Aug 2025).

Optimizations include:

• Quadratic polynomial approximation for gain updates ($g = Au^2 + Bu + C$) using pre-calculated lookup tables.
• Pipelined AGC and filter operations (e.g., 100 pipeline stages to support throughput).
• Fixed-point quantization for internal and I/O signals (18–24 bits).
• BlockRAM-based coefficient storage for time-multiplexed filter reuse.

These design choices result in hardware utilization as low as 13.5% for a 64-channel instance and theoretical scalability to parallel multi-instance deployment.

4. Event-Driven and Strided Feature Extraction in Neuromorphic Systems

The spike-based outputs of the CARFAC model have been paired with event-driven neuromorphic feature extraction techniques. One prominent methodology is FEAST (Feature Extraction using Adaptive Selection Thresholds), which processes leaky integrate-and-fire (LIF) neuron outputs derived from CARFAC filter banks (Xu et al., 2022).

In FEAST, asynchronous events (spikes) are mapped into event-context vectors with exponential decay kernels:

$$S_i(u, p) = \exp\left(-\frac{t - l_i(u, p)}{T_y}\right)$$

Feature learning is competitive and uses cosine similarity for neuron activation.

• 1D FEAST operates on temporal event contexts in a single channel.
• 2D FEAST analyzes joint spectro-temporal contexts across multiple frequency channels, leveraging the underlying tonotopic mapping of CARFAC (inspired by the Greenwood function).

Empirical results demonstrate high spoken digit classification accuracy (TIDIGITS: up to ~97.7%) with relatively low-complexity FEAST-based networks, rivaling deep neural network approaches with considerably lower power and computational cost.

5. Software Advances and Model Extensions

CARFAC has matured into a cross-platform, open-source solution, with notable enhancements in version 2 ("CARFAC v2") (Lyon et al., 26 Apr 2024):

• Matlab fidelity improvements (bug fixes, AC-coupling relocation, gain update refinements).
• New implementations in Python/NumPy and JAX for high-performance and differentiable simulation, sharing a unified test suite for validation.
• The JAX version introduces pytrees for distinguishing parameters and state, enabling gradient-based optimization and seamless integration with modern ML workflows.

Algorithmic updates in v2:

• DC quadratic distortion anomaly addressed by relocating the 20 Hz highpass AC-coupling filter from the IHC block into the CAR cascade, thus eliminating unnatural persistent DC output.
• Two-capacitor IHC model reduces neural synchrony at high frequencies by implementing sequential LPFs (200 μs and 80 μs time constants).
• Parameterization for hearing loss simulation via an "ohc_health" vector, enabling differentiated channel-level amplifier function modeling.

Additionally, integration with the Auditory Model Toolbox (AMT) has been extensively improved, enhancing interoperability and supporting multi-model analysis in computational auditory research.

6. Comparative Context and Physical Model Alternatives

While CARFAC is fundamentally phenomenological—with its core mechanics and nonlinearities engineered to reproduce cochlear auditory filtering, frequency selectivity, and compression—contrasting first-principles physical models have been pursued.

A notable example is the fluid-coupled array of subwavelength Hopf resonators, which naturally reproduces tonotopic separation and nonlinear amplification via direct modal decomposition and Hopf bifurcation dynamics (Ammari et al., 2019). This physical approach:

• Maps exponential grading in resonator sizes to frequency separation,
• Implements nonlinearity at the pressure velocity level,
• Retains full spatial coupling among resonators, offering a physically grounded alternative to CARFAC’s cascaded filter—and engineering analogues.

Both CARFAC and Hopf-resonator approaches aim to reproduce sharp cochlear tuning with compressive gain control, although CARFAC is more widely adopted for neuromorphic and embedded applications owing to its efficiency and direct mapping to biological cochlear mechanics.

7. Applications and Research Significance

CARFAC’s scalable, real-time implementation supports a breadth of machine-hearing tasks:

• Front-end spectral decomposition for speech and audio recognition (FPGA-based cochlear front-ends).
• Neuromorphic systems for low-latency acoustic sensing, including underwater environments with stringent energy and throughput requirements (Bremer et al., 11 Aug 2025).
• Hearing impairment simulation and comparative auditory model studies (parameterized amplifier loss, multi-model toolbox integration).

Its ongoing refinement (e.g., quadratic approximations for hardware resource constraints, differentiable implementations for ML workflows) ensures its relevance both for computational neuroscience research and for deployment in embedded sensory systems.

---

A plausible implication is that CARFAC will continue to serve as the canonical cochlear model for both bio-inspired signal processing and algorithmic exploration in neuromorphic and auditory science fields, with future work likely to further bridge biological fidelity and computational efficiency.