ETD

• dendrites
• dendritic nonlinearities
• dendritic subunits
• dynamical systems
• dynamics
• fixed points
• latent variables
• neural networks
• neurons
• recurrent neural networks
• trajectories

A common approach to studying neural computations is the state-space framework, in which the collective activity of a neural network is represented as a vector in a high-dimensional state space. Several studies have shown that neural trajectories remain confined to low-dimensional manifolds within this space. An explanation for how neuronal connectivity gives rise to such highly organized collective activity was proposed by Mastrogiuseppe and Ostojic, who modeled brain circuits as low-rank recurrent neural networks (RNNs) characterized by low-rank connectivity matrices. The dynamics and fixed-point structure of these networks are now well understood thanks to the work of Mastrogiuseppe, Ostojic, and collaborators, who exploited the low-dimensional dynamical systems governing a small number of latent variables to infer the properties of the corresponding high-dimensional trajectories.
However, these models rely on the classical point-neuron approximation, which neglects the nonlinear integration processes occurring within dendrites. Experimental studies by Polsky and colleagues provided evidence for a two-stage model of synaptic integration in pyramidal neurons, suggesting that dendritic nonlinearities may play an important computational role.
The central question addressed in this study is how the dynamics of large low-rank RNNs are modified when neurons are modeled as two-stage units incorporating dendritic nonlinearities.
Our results show that the network fixed point undergoes a qualitative transition as the strength of the external drive increases. For weak stimuli, the fixed point depends on the geometric parameters of the network. In contrast, for strong inputs, it approaches a configuration determined solely by the external-input weight vector. In particular, for weak external inputs, both the network dynamics and the fixed-point configuration coincide with those of the linear-dendrite model. For sufficiently strong drives, however, the fixed point deviates significantly from its linear counterpart.
The threshold separating these two regimes is determined by analyzing the variance of the input received by a generic dendritic subunit. We find that this threshold grows linearly with the number of dendrites and depends on the geometric parameters characterizing the network. The same analysis is extended to networks whose neurons combine nonlinear dendritic processing with sigmoidal somatic transfer functions. The results reveal qualitatively similar phenomena, indicating that the transition between geometry-dependent and geometry-independent fixed-point regimes is robust to the introduction of somatic sigmoid nonlinearities.
Overall, these findings demonstrate that dendritic nonlinearities can qualitatively reshape network dynamics and should not be regarded merely as a biophysical detail.