Neural Manifolds and Cognitive Consistency: A New Approach to Memory Consolidation in Artificial Systems (2503.01867v1)

Abstract: We introduce a novel mathematical framework that unifies neural population dynamics, hippocampal sharp wave-ripple (SpWR) generation, and cognitive consistency constraints inspired by Heider's theory. Our model leverages low-dimensional manifold representations to capture structured neural drift and incorporates a balance energy function to enforce coherent synaptic interactions, effectively simulating the memory consolidation processes observed in biological systems. Simulation results demonstrate that our approach not only reproduces key features of SpWR events but also enhances network interpretability. This work paves the way for scalable neuromorphic architectures that bridge neuroscience and artificial intelligence, offering more robust and adaptive learning mechanisms for future intelligent systems.

• The paper proposes a unified mathematical framework representing neural population activity on a low-dimensional manifold and modeling memory replay (SpWR) as diffusion processes.
• It integrates cognitive consistency constraints from Heider's theory via a balance energy function, enhancing memory stability and network interpretability.
• Algorithmic innovations and simulations show that applying balance constraints improves deep learning model interpretability without losing predictive performance.

The paper proposes a comprehensive mathematical framework that integrates neural population dynamics, hippocampal sharp wave-ripple (SpWR Sharp Wave-Ripple) generation, and cognitive consistency constraints inspired by Heider’s consistency theory. The main contributions can be summarized as follows:

• Unified Modeling of Neural Dynamics and Memory Replay:

The framework formalizes the evolution of neural population activity using a low-dimensional manifold representation. Neural firing rates, denoted by $\mathbf{r}_t$, are modeled as a function of an agent’s position in an environment and additive Gaussian noise. In particular, the state space is constrained on a manifold $$\mathcal{M} = \{ \mathbf{r}(\mathbf{x}, k) \mid \mathbf{x}\in \text{maze}, \; k\in\{1,2,\dots,K\} \},$$ which captures the structured drift of neural representations. SpWR events are modeled as diffusion processes on this manifold. The mapping between the neural activity during wakefulness and replay events is achieved through probabilistic mechanisms that compare instantaneous SpWR patterns with stored trial block templates via similarity measures (e.g., cosine similarity) and an inverse temperature parameter $\beta$.

• Memory Consolidation via Synaptic Plasticity and Replay Bias:

The framework models memory consolidation by leveraging both Hebbian synaptic plasticity and a probabilistic selection of trial blocks. The weight update rule is defined as $$\Delta w_{ij}\propto \eta \sum_{m} P(k|m) \, \mathbf{s}_m^{(i)}\mathbf{s}_m^{(j)},$$ where $\eta$ is the learning rate and $P(k|m)$ biases replay toward trial blocks with salient activity profiles. Importantly, the probabilistic replay during both wakefulness and sleep incorporates an amplification factor $\gamma$, ensuring that frequently activated experiences during active states are even more likely to be consolidated during offline replay.

• Integration of Cognitive Consistency through Balance Energy:

Drawing from Heider’s consistency theory, the paper introduces a balance energy function defined over a signed graph of neural interactions. Neurons are connected via weights $W_{ij}$ that can be either excitatory or inhibitory, and triads of neurons are assessed according to the product $W_{ij}W_{jk}W_{ki},$ where balanced triads (positive product) are energetically favorable. The corresponding energy function is given by $$E_{\text{balance}} = \sum_{i<j<k} \left(W_{ij}W_{jk}W_{ki} + 1\right)^2.$$ Neural population activity is then modified by an additional gradient term, $-\alpha\nabla_{\mathbf{r}}E_{\text{balance}}$, which enforces structured, balanced interactions. This mechanism not only stabilizes memory representations but also enhances network interpretability by promoting attractor dynamics in low-energy (i.e., high-consistency) regions of the manifold.

• Efficient Algorithmic Implementations:

To address computational challenges, the paper derives vectorized expressions for both energy and gradient computations by relating triple summations over triads to matrix traces. For instance, it is shown that for a symmetric weight matrix, $$\sum_{i<j<k}W_{ij}W_{jk}W_{ki} = \frac{1}{6}\operatorname{tr}(W^3),$$ leading to a fast formulation of the balance energy as $$E^\text{fast}_{\text{balance}} = \binom{N}{3} + \frac{1}{3}\operatorname{tr}(W^3) + \frac{1}{6} \operatorname{tr}((W\circ W)^3).$$ The corresponding gradient is computed as $$\text{grad\_fast} = 2\Bigl(W^2 + \bigl[(W\circ W)^2 \circ W\bigr]\Bigr).$$ These developments are crucial for scaling the framework to large neural ensembles and neuromorphic implementations.

• Multi-Expert Modeling and Neural Interpretability:

Beyond theoretical development, the paper extends the framework to interpret deep learning models such as recurrent neural networks (RNNs), long short-term memory networks (LSTM), and gated recurrent units (GRU). Knowledge embeddings are mapped onto fixed neuron embeddings via an embedding function, and a clustering approach (e.g., using $K$-Means) organizes the neural representation. Synaptic weight optimization through gradient descent (with empirically chosen parameters, such as $\eta=0.001$ for the balance dynamics and $200$ iterations) further demonstrates that imposing balance constraints can improve the interpretability of neural connections without degrading classification performance. The results indicate that across different architectures, the balance energy consistently decreases over training epochs while preserving the decision boundary, suggesting that the approach reinforces network robustness and selective synaptic strengthening.

• Empirical Evaluations and Simulation Results:
  • Low-dimensional embedding trajectories capture the diffusion-driven evolution of neural states during SpWR events.
  • Similarity heatmaps between replay events and stored trial blocks demonstrate selective activation.
  • Comparative plots of trial block activation frequencies during wakefulness and sleep validate the power-law selection mechanism driven by $\gamma$.
  • Analysis of synaptic weight distributions reveals a long-tailed pattern consistent with Hebbian predictiveness.
  • Visualizations of neural network interpretations (using RNN, LSTM, and GRU models) underscore that balance regularization enhances interpretability while maintaining predictive performance.

In summary, the paper elegantly bridges neurobiological processes with computational modeling by unifying low-dimensional neural manifolds, probabilistic memory replay, and balance-driven synaptic plasticity. The integration of Heider’s consistency theory into the modeling of synaptic interactions provides a novel constraint that promotes stable, interpretable memory consolidation, with several strong numerical experiments and algorithmic innovations underpinning its claims.