Phase state space models train spiking networks without surrogates

What the work claims

Olin-Ammentorp introduces Phase State Space Models, or phi-SSMs, as a way to train spiking neural networks built from resonate-and-fire neurons.¹ The core claim is that by fixing the resonant frequency of each layer, the relative phases of neurons become time-invariant and can be decoded continuously. That lets the network produce gradients at every time step, even for neurons that do not spike, without the smoothed spike proxies normally used in surrogate gradient learning. The same formulation supports three equivalent execution modes: a continuous differential update suited to physical oscillators, a discrete iterative update for digital cores, and a convolutional or fast-Fourier-transform view for parallel training. The author also implements an STFT adapter and an attention module that operate entirely in the phase domain, and shows a formal equivalence to Fourier Holographic Reduced Representations, a hyperdimensional computing scheme.

How it works

A resonate-and-fire neuron is defined by a complex-valued membrane potential U that evolves according to dU/dt = k U + w I, where the eigenvalue k = lambda + i omega has a negative damping term lambda and an angular frequency omega. The neuron fires when the magnitude of U crosses a threshold while its phase passes through zero. In the phi-SSM, the output is not the full complex potential but the instantaneous relative phase, obtained by demodulating U with a reference oscillator at the layer's fixed frequency.

Because the relative phase is continuous, its derivative is well-defined and standard automatic differentiation can be applied. The discrete update samples the potential at the resonant period and produces a convolutional kernel K[n] = exp(k n T); taking the FFT lets the network compute all states across time in parallel during training, then execute recurrently at inference. The author shows numerically that the continuous, discrete and FFT views give identical outputs.

To handle inputs that are not already phase-coded, an STFT adapter uses trainable-frequency resonate-and-fire compartments to extract frequency components, then demodulates them onto the common layer frequency. An attention module projects phase inputs into queries, keys and values using three phi-SSM layers and computes scores with the hyperdimensional cosine-similarity operator. A codebook readout maps the final phase vector to a class by maximizing similarity with stored phase codes.

Where a skeptic should push

The empirical demonstration is thin. The paper shows sequential FashionMNIST and a selective copying task, but provides no detailed accuracy numbers, sample sizes or error analyses in the main text. The attention module is described as necessary for the copying task to exceed chance and the STFT adapter is described as allowing full performance, but these claims are qualitative. Without quantitative results, it is hard to judge whether the phase representation scales beyond toy sequence problems.

The frequency-locking assumption is also strong. Fixing omega across a layer makes relative phases invariant, which is mathematically elegant, but it removes the ability of individual neurons to resonate at different frequencies. The STFT adapter reintroduces frequency diversity at the input, but the rest of the network operates in a single phase band. Whether that is sufficient for complex temporal pattern separation is an open question.

Finally, the work is purely algorithmic. There are no neuromorphic hardware results, no energy measurements and no comparison to state-of-the-art SNN training methods on standard benchmarks such as SHD or DVS Gesture. The bridge to hyperdimensional computing is conceptually clean, but its practical utility remains to be shown.

What this changes for modeling organoid dynamics

The non-obvious implication for organoid intelligence is that the paper supplies a mathematical language for the oscillatory, subthreshold dynamics that organoid recordings actually exhibit. Standard leaky integrate-and-fire models discard most of the membrane dynamics between spikes. Resonate-and-fire neurons, by contrast, naturally capture resonance, bursting and phase relationships, which are prominent in developing neural cultures. The phi-SSM formulation means one can define loss functions and compute gradients on those richer dynamics without waiting for a spike.

The opportunity is twofold. First, phi-SSMs could serve as an in-silico model of an organoid, helping to decode sparse and noisy multielectrode array data. Because gradients are available from subthreshold phase, the model could be fit to recordings in which most neurons spike rarely, a common situation in organoids. Second, the phase code offers a compact way to translate between external stimuli and organoid activity: a stimulus could be encoded as relative phases, delivered through patterned electrical or optogenetic stimulation, and the organoid's phase response could be read out without binning spikes.

The threat is over-idealization. Organoid networks are heterogeneous, drift over time, and do not respect a single global frequency. The phi-SSM assumes a fixed resonant frequency and clean phase relationships; applying it directly to organoid data may impose structure that is not there. If researchers treat the phase model as a ground-truth description of biological computation rather than a useful approximation, they risk mistaking model artifacts for neural signals. The hype-correction is that richer neuron models help, but they do not by themselves solve the harder problems of reproducibility, scaling and long-term stability in biological substrates.

On the dual-use side, phase-coded spiking networks are attractive for ultra-low-power sequence processing. If organoids can be made to implement such codes, the same capabilities could be redirected toward surveillance or autonomous decision systems, raising the usual concerns about computing on living neural tissue.

The bottom line

This is a theoretical and algorithmic contribution: a surrogate-free training framework for resonate-and-fire networks with a clean connection to state-space models and hyperdimensional computing. The empirical evidence is limited to small sequence tasks, so claims about scalability should be treated as hypotheses. For organoid intelligence, the value lies in the richer dynamical vocabulary it provides, not in any immediate experimental applicability. It points toward models that take phase and subthreshold resonance seriously, which is where organoid research may eventually need to go.