Dendritic LMS State for Spiking ICL

Integrate this at the hidden-state level of a small recurrent SNN or a single-layer spiking token processor. Encode each scalar or vector input example into a feature vector $x_t\in\mathbb{R}^d$ using a fixed random projection or a learned linear encoder. Maintain one persistent floating-point apical tensor $a_t\in\mathbb{R}^d$ per sequence and initialize it to zero at the beginning of every task. For a labeled context pair $(x_t,y_t)$, compute the prediction $\hat y_t=a_t^\top x_t$, the error $e_t=y_t-\hat y_t$, and update the apical state with $a\leftarrow\lambda a+\eta e_t x_t$. Feed $a$ plus the current input through a learned soma/readout layer, apply a spiking nonlinearity such as an LIF threshold, and use surrogate gradients only for the encoder, readout, and optional spike parameters. On the query token, compute $\hat y=a^\top x_{query}$ and freeze $a$; do not accidentally train or update it from the query loss during the forward pass. A compact pseudocode loop is: a=zeros(d); for (x,y,query_flag) in sequence: z=encoder(x); pred=a@z; if not query_flag: err=y-pred; a=lambda*a+eta*err*z; spikes=surrogate_soma(a,z); output=readout(spikes,pred). The mathematical quantities $\lambda$, $\eta$, the LMS error, and the outer-product update are computed exactly; feature normalization, spike thresholds, and the encoder are learned or estimated empirically. Start with the scalar linear Garg-2022 protocol at $d=20,50,100$, compare against a Pure LIF network and a similarly sized GRU or Transformer, and run at least 20 random seeds. Success means higher query $R^2$, substantially lower seed variance, and performance that remains stable as task dimension exceeds the baseline's grokking or collapse regime. Also record state size, synaptic operations, and wall-clock time to test whether the single-layer state gives an inference-memory or latency advantage.