Field-theoretic approach to compartmental neuronal networks: impact of dendritic calcium spike-dependent bursting

Neurons are spatially extended cells; different parts of a neuron have specific voltage dynamics. Important types of neurons even generate different spikes in different parts of the cell. Neurons’ inputs are also often spatially compartmentalized, with different sources targeting different locations on the cell. Classic mean-field theories for neural population activity, however, rely on point-neuron models with at most one type of spike. Here, we develop a statistical field-theoretic approach to understanding collective activity in networks of compartmental neurons, including those generating multiple types of spikes. We use this to examine simple models of networks with thick-tufted layer 5 pyramidal cells, which generate calcium spikes in their apical dendrite when dendritic depolarization coincides with a back-propagating somatic spike. In the weakly-coupled regime, we uncover an exact mean-field limit for these networks that maps them to a marked point process. We use this mean-field limit to compare the impact of compartmentalized recurrent excitatory and inhibitory connectivity on the equilibrium phase diagram. This exposes regions of metastability between various activity states, including activity with silent vs active dendrites, with and without inhibitory activity, and oscillations.

💡 Research Summary

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The paper introduces a statistical field‑theoretic framework for neural networks composed of compartmental neurons, specifically targeting thick‑tufted layer 5 pyramidal cells that generate both somatic Na⁺/K⁺ spikes and dendritic calcium spikes. Traditional mean‑field approaches treat neurons as point‑like units with a single spike type, which cannot capture the spatial segregation of inputs and the interaction between distinct spike mechanisms. To overcome this limitation, the authors model each neuron as a set of isopotential compartments (soma S and apical dendrite D). The subthreshold voltage dynamics of each compartment follow a linear differential equation driven by external inputs and synaptic currents, while spike generation is represented by point processes whose intensities depend on the local voltage through transfer functions: a rectified power‑law (or threshold‑linear) function (f(v)) for somatic spikes and a bounded probability function (g(v)) for dendritic calcium spikes that are triggered only when a somatic spike back‑propagates into a sufficiently depolarized dendrite.

In the weak‑coupling limit (synaptic weights scale as (N^{-1})), the authors derive an exact mean‑field limit. The network activity collapses onto a marked point process: each dendritic calcium spike marks the first spike of a burst, while subsequent spikes in the burst are unmarked. This mapping yields closed self‑consistent equations (Eq. 8) for the population‑averaged firing rates of somatic and dendritic spikes, (\langle\dot a_S\rangle) and (\langle\dot a_D\rangle). The equations incorporate the linear voltage filters (G_S) and (G_D), the external drives (E_S) and (E_D), and the mean synaptic kernels for soma‑targeted ((J_S)) and dendrite‑targeted ((J_D)) connections, with a scaling factor (\beta) that captures the extra postsynaptic effect of a dendritic burst.

Stability analysis is performed by averaging the voltage dynamics (Eq. 9), leading to coupled differential equations for the mean voltages (\langle v_S\rangle) and (\langle v_D\rangle). Fixed points of these equations correspond to stationary firing‑rate solutions of the marked point process. Two connectivity scenarios are examined:

1. Soma‑targeted recurrent connectivity (J_D = 0). The mean‑field reduces to a one‑dimensional equation for (\langle\dot a_S\rangle). Three regimes emerge: (a) a silent state (no spikes), (b) somatic spiking without dendritic calcium bursts, and (c) spiking with active dendrites (either sub‑saturated or fully saturated when (g=1)). If the effective recurrent gain (J(1+\beta g(E_D))) exceeds unity, the system exhibits a runaway excitation leading to divergence of firing rates. The phase diagram in the ((E_S,E_D)) plane shows clear boundaries between these regimes, confirmed by network simulations.

2. Dendrite‑targeted recurrent connectivity (J_S = 0). Here the soma receives only external drive, while dendritic activity is amplified by recurrent excitation. The mean‑field again yields three fixed points, but a qualitatively different behavior appears: strong dendritic recurrence can generate bistability between a state with silent dendrites and a state with active dendritic bursts, even when somatic firing is unchanged. This bistability is absent in the soma‑targeted case and reflects the conditional nature of calcium spikes.

The authors illustrate these regimes with raster plots, rate curves, and analytical phase diagrams, demonstrating that the field‑theoretic predictions match stochastic simulations across parameter ranges. They also discuss how the model captures biologically relevant motifs: SOM‑Martinotti interneurons preferentially inhibit dendrites, while PV‑basket cells target somata. The framework therefore provides a quantitative tool to explore how compartment‑specific inhibition shapes network dynamics, potentially influencing attentional gating, working memory, and pathological states such as seizures.

In conclusion, the paper delivers a rigorous statistical mechanics description of compartmental neuronal networks, extending mean‑field theory to accommodate multiple spike types and spatially segregated synaptic targeting. The exact mapping to a marked point process and the derived stability conditions open avenues for studying learning‑induced plasticity, non‑linear voltage propagation, and disease modeling within a tractable analytical setting. Future work could incorporate non‑linear dendritic conductances, activity‑dependent synaptic plasticity, and empirical validation against in‑vivo calcium imaging or electrophysiology data.