arXiv:2609.12327v1 Announce Type: new Abstract: Reservoir computing (RC) trains only a linear readout over a fixed recurrent layer, making it fast and data-efficient for online prediction. However, a static reservoir degrades under system drift, readout-only adaptation is then insufficient, and unconstrained reservoir adaptation can destroy the echo-state and incremental stability properties that make RC reliable. This paper proposes LoRA-RC, which adapts the recurrent matrix through a low-rank correction driven by streaming prediction errors. The base reservoir and adaptation bases are fixed offline; a small core matrix is adapted online, projected onto a spectral-norm ball, and low-pass filtered at each step. The projection guarantees that every applied recurrent matrix remains within a certified contraction set, and an incremental input-to-state stability bound is established for the reservoir along each online adaptation path, with path-independent rate and gain. On a Lorenz system with an abrupt parameter drift, LoRA-RC cuts post-drift prediction error by 56% versus a fixed RC and 51% versus readout-only adaptation; ablations over 20 seeds show that removing the projection inflates this error by more than a factor of 40.

arXiv:2609.13074v1 Announce Type: cross Abstract: The maintenance of continuous variable information in working memory is thought to rely on persistent patterns of cortical activity. In delayed-estimation tasks, neural activity can form localized activity peaks, or ``bumps,'' whose positions track the remembered variable. Such activity is well described by continuous-attractor neural field models, but most existing models collapse cortical inhibition into a single homogeneous population. Here, we introduce a stochastic neural field model with distinct excitatory, parvalbumin-expressing (PV), and somatostatin-expressing (SST) populations to examine how inhibitory subtype structure shapes persistent activity. Using a Heaviside firing-rate approximation, we derive stationary bump solutions and reduce their linear stability to separate shifting and scaling modes. We show that population thresholds and inhibitory timescales determine both bump stability and the mechanism by which stability is lost, while inhibitory connection strengths and spatial scales substantially reshape the stable parameter region. In particular, broader SST connectivity promotes stable bump states. Finally, we derive an effective diffusion coefficient for noise-driven bump wandering and show that increasing the SST spatial footprint reduces the rate of memory diffusion. Together, these results demonstrate how inhibitory subtype structure can shape both the deterministic stability and stochastic precision of continuous-attractor memories.

arXiv:2602.08640v5 Announce Type: replace-cross Abstract: Universal approximation theorems establish the expressive capacity of neural network architectures. For dynamical systems, existing results are limited to finite time horizons or systems with a globally stable equilibrium, leaving multistability and limit cycles unaddressed. We prove that Neural ODEs achieve $\varepsilon$-$\delta$ closeness -- trajectories within error $\varepsilon$ except for initial conditions of measure $

arXiv:2603.11066v5 Announce Type: replace-cross Abstract: We develop a structural framework for the Collatz map based on odd-to-odd dynamics, modular return structure, and burst-gap decomposition. We prove exact results for fiber-57 return dynamics, including a uniform affine channel for q = 7 (mod 8), minimum return gap five for q = 3 (mod 8), and that the chain map on the invariant core is a permutation at every depth. These yield a depth-2 partial return kernel with Perron root 129/1024. A conditional reduction via phantom-cycle gain analysis and a weak-mixing hierarchy establishes an exact geometric block law, an exponential almost-all crossing bound, and per-orbit phantom gain within a 4.65x contraction margin, reducing convergence to a single orbitwise regularity statement. New in v5: the Carry Contamination Theorem proves that at every Sturmian depth D, n_D mod 8 is exactly equidistributed over {1,3,5,7}, yielding an exact (3/4)^D survivor law. The reduction terminates at the Carry Independence Conjecture (CIC): no n_0 > 1 avoids class 5 at every depth. CIC implies Collatz via a proved Dichotomy theorem. This extends via a seven-block cross-core alphabet with spectral radius rho = 2+sqrt(2). The safe Collatz map is a 2-adic expander with measure shrinkage mu_2(T_j) = 5 non-descending rounds for large K. Cycle impossibility is verified through period 13 (238,811 words). Anti-correlation is bounded (constant ~0.635, not exponential in K). This is not a proof of the Collatz conjecture but a sharpened structural reduction along two routes, each terminating at a distributional-to-pointwise gap.

arXiv:2412.04172v3 Announce Type: replace Abstract: Neurons rely on two interdependent mechanisms, homeostasis and neuromodulation, to maintain robust and adaptable functionality. Calcium homeostasis stabilizes neuronal activity by adjusting ionic conductances, whereas neuromodulation dynamically modifies ionic properties in response to external signals carried by neuromodulators. Combining these mechanisms in conductance-based models often produces unreliable outcomes, particularly when sharp neuromodulation interferes with calcium-homeostatic tuning. This study explores how a biologically inspired neuromodulation controller can harmonize with calcium homeostasis to ensure reliable neuronal function. Using computational models of stomatogastric ganglion and dopaminergic neurons, we demonstrate that controlled neuromodulation preserves neuronal firing patterns while calcium homeostasis simultaneously maintains target intracellular calcium levels. Unlike sharp neuromodulation, the neuromodulation controller integrates activity-dependent feedback through mechanisms mimicking G-protein-coupled receptor cascades. The interaction between these controllers critically depends on the existence of an intersection in conductance space, representing a balance between target calcium levels and neuromodulated firing patterns. Maximizing neuronal degeneracy enhances the likelihood of such intersections, enabling robust modulation and compensation for channel blockades. We further show that this controller pairing extends to network-level activity, reliably modulating the rhythmic activity of central pattern generators. This study highlights the complementary roles of calcium homeostasis and neuromodulation, proposing a unified control framework for maintaining robust and adaptive neural activity under physiological and pathological conditions.

arXiv:2604.00277v2 Announce Type: replace-cross Abstract: Energy-based models (EBMs) implement inference as gradient descent on a learned Lyapunov function, yielding interpretable, structure-preserving alternatives to black-box neural ODEs and aligning naturally with physical AI. Yet their use in system identification remains limited, and existing architectures lack formal stability guarantees that globally preclude unstable modes. We address this gap by introducing an EBM framework for system identification with stable, dissipative, absorbing invariant dynamics. Unlike classical global Lyapunov stability, absorbing invariance expands the class of stability-preserving architectures, enabling more flexible and expressive EBMs. We extend EBM theory to nonsmooth activations by establishing negative energy dissipation via Clarke derivatives and deriving new conditions for radial unboundedness, exposing a stability-expressivity tradeoff in standard EBMs. To overcome this, we introduce a hybrid architecture with a dynamical visible layer and static hidden layers, prove absorbing invariance under mild assumptions, and show that these guarantees extend to port-Hamiltonian EBMs. Experiments on metric-deformed multi-well and ring systems validate the approach, showcasing how our hybrid EBM architecture combines expressivity with sound and provable safety guarantees by design.

arXiv:2603.22871v1 Announce Type: new Abstract: Current large language models (LLMs) primarily rely on linear sequence generation and massive parameter counts, yet they severely struggle with complex algorithmic reasoning. While recent reasoning architectures, such as the Hierarchical Reasoning Model (HRM) and Tiny Recursive Model (TRM), demonstrate that compact recursive networks can tackle these tasks, their training dynamics often lack rigorous mathematical guarantees, leading to instability and representational collapse. We propose the Contraction Mapping Model (CMM), a novel architecture that reformulates discrete recursive reasoning into continuous Neural Ordinary and Stochastic Differential Equations (NODEs/NSDEs). By explicitly enforcing the convergence of the latent phase point to a stable equilibrium state and mitigating feature collapse with a hyperspherical repulsion loss, the CMM provides a mathematically grounded and highly stable reasoning engine. On the Sudoku-Extreme benchmark, a 5M-parameter CMM achieves a state-of-the-art accuracy of 93.7 %, outperforming the 27M-parameter HRM (55.0 %) and 5M-parameter TRM (87.4 %). Remarkably, even when aggressively compressed to an ultra-tiny footprint of just 0.26M parameters, the CMM retains robust predictive power, achieving 85.4 % on Sudoku-Extreme and 82.2 % on the Maze benchmark. These results establish a new frontier for extreme parameter efficiency, proving that mathematically rigorous latent dynamics can effectively replace brute-force scaling in artificial reasoning.

arXiv:2603.12878v1 Announce Type: new Abstract: The synchronized activity of neuronal populations can lead to pathological over-synchronization in conditions such as epilepsy and Parkinson disease. Such states can be desynchronized by brief electrical pulses. But when the underlying oscillating system is not known, as in most practical applications, to determine the specific times and intensities of pulses used for desynchronizaton is a difficult inverse problem. Here we propose a desynchronization scheme for neuronal models of bi-variate neural activity, with possible applications in the medical setting. Our main argument is the existence of a peculiar point in the phase space of the system, the centroid, that is both easy to calculate and robust under changes in the coupling constant. This important target point can be used in a control procedure because it lies in the region of minimal return times of the system.

arXiv:2509.12783v2 Announce Type: replace Abstract: Inferring the biophysical parameters of conductance-based models (CBMs) from experimentally accessible recordings remains a central challenge in computational neuroscience. Spike times are the most widely available data, yet they reveal little about which combinations of ion channel conductances generate the observed activity. This inverse problem is further complicated by neuronal degeneracy, where multiple distinct conductance sets yield similar spiking patterns. We introduce a method that addresses this challenge by combining deep learning with Dynamic Input Conductances (DICs), a theoretical framework that reduces complex CBMs to three interpretable feedback components governing excitability and firing patterns. Our approach first maps spike times to DIC densities at threshold using a neural network that learns a low-dimensional representation of neuronal activity. The predicted DIC values are then used to generate degenerate CBM populations via an iterative compensation algorithm, ensuring compatibility with the intermediate target DICs, and thereby reproducing the corresponding firing patterns, even in high-dimensional models. Applied to two models, this algorithmic pipeline reconstructs spiking and bursting regimes with high accuracy and robustness to variability, including spike trains generated under noisy current injection mimicking physiological stochasticity. It produces diverse degenerate populations within milliseconds on standard hardware, enabling scalable and efficient inference from spike recordings alone. Together, this work positions DICs as a practical and interpretable link between experimentally observed activity and mechanistic models. By enabling fast and scalable reconstruction of degenerate populations directly from spike times, our approach provides a powerful way to investigate how neurons exploit conductance variability to achieve reliable computation.

arXiv:2603.03201v1 Announce Type: cross Abstract: Reasoning is the ability to integrate internal states and external inputs in a meaningful and semantically consistent flow. Contemporary machine learning (ML) systems increasingly rely on such sequential reasoning, from language understanding to multi-modal generation, often operating over dictionaries of prototypical patterns reminiscent of associative memory models. Understanding retrieval and sequentiality in associative memory models provides a powerful bridge to gain insight into ML reasoning. While the static retrieval properties of associative memory models are well understood, the theoretical foundations of sequential retrieval and multi-memory integration remain limited, with existing studies largely relying on numerical evidence. This work develops a dynamical theory of sequential reasoning in Hopfield networks. We consider the recently proposed input-driven plasticity (IDP) Hopfield network and analyze a two-timescale architecture coupling fast associative retrieval with slow reasoning dynamics. We derive explicit conditions for self-sustained memory transitions, including gain thresholds, escape times, and collapse regimes. Together, these results provide a principled mathematical account of sequentiality in associative memory models, bridging classical Hopfield dynamics and modern reasoning architectures.

arXiv:2012.15834v3 Announce Type: replace-cross Abstract: Neural network training is commonly based on SGD. However, the understanding of SGD's ability to converge to good local minima, given the non-convex nature of loss functions and the intricate geometric characteristics of loss landscapes, remains limited. In this paper, we apply topological data analysis methods to loss landscapes to gain insights into the learning process and generalization properties of deep neural networks. We use the loss function topology to relate the local behavior of gradient descent trajectories with the global properties of the loss surface. For this purpose, we define the neural network's Topological Obstructions score ("TO-score") with the help of robust topological invariants, barcodes of the loss function, which quantify the escapability of local minima for gradient-based optimization. Our two principal observations are: 1) the loss barcode of the neural network decreases with increasing depth and width, therefore the topological obstructions to learning diminish; 2) in certain situations there is a connection between the length of minima segments in the loss barcode and the minima's generalization errors. Our statements are based on extensive experiments with fully connected, convolutional, and transformer architectures and several datasets including MNIST, FMNIST, CIFAR10, CIFAR100, SVHN, and multilingual OSCAR text dataset.