Dendrites of pyramidal cells exhibit complex morphologies and contain a variety of ionic conductances, which generate non-trivial integrative properties. Basal and proximal apical dendrites have been shown to function as independent computational subunits within a two-layer feedforward processing scheme. The outputs of the subunits are linearly summed and passed through a final non-linearity. It is an open question whether this mathematical abstraction can be applied to apical tuft dendrites as well. Using a detailed compartmental model of CA1 pyramidal neurons and a novel theoretical framework based on iso-response methods, we first show that somatic sub-threshold responses to brief synaptic inputs cannot be described by a two-layer feedforward model. Then, we relax the core assumption of subunit independence and introduce non-linear feedback from the output layer to the subunit inputs. We find that additive feedback alone explains the somatic responses to synaptic inputs to most of the branches in the apical tuft. Individual dendritic branches bidirectionally modulate the thresholds of their input-output curves without significantly changing the gains. In contrast to these findings for precisely timed inputs, we show that neuronal computations based on firing rates can be accurately described by purely feedforward two-layer models. Our findings support the view that dendrites of pyramidal neurons possess non-linear analog processing capabilities that critically depend on the location of synaptic inputs. The iso-response framework proposed in this computational study is highly efficient and could be directly applied to biological neurons.

Dendritic spines are submicron, subcellular compartments whose shape is defined by actin filaments and associated proteins. Accurately mapping the cytoskeleton is a challenge, given the small size of its components. It remains unclear whether the actin-associated structures analyzed in dendritic spines of neurons in vitro apply to dendritic spines of intact, mature neurons in situ. Here, we combined advanced preparative methods with multitilt serial section electron microscopy (EM) tomography and computational analysis to reveal the full three-dimensional (3D) internal architecture of spines in the intact brains of male mice at nanometer resolution. We compared hippocampal (CA1) pyramidal cells and cerebellar Purkinje cells in terms of the length distribution and connectivity of filaments, their branching-angles and absolute orientations, and the elementary loops formed by the network. Despite differences in shape and size across spines and between spine heads and necks, the internal organization was remarkably similar in both neuron types and largely homogeneous throughout the spine volume. In the tortuous mesh of highly branched and interconnected filaments, branches exhibited no preferred orientation except in the immediate vicinity of the cell membrane. We found that new filaments preferentially split off from the convex side of a bending filament, consistent with the behavior of Arp2/3-mediated branching of actin under mechanical deformation. Based on the quantitative analysis, the spine cytoskeleton is likely subject to considerable mechanical force in situ .

The cable equation is key for understanding the electrical potential along dendrites or axons, but its application to dendritic spines remains limited. Their volume is extremely small so that moderate ionic currents suffice to alter ionic concentrations. The resulting chemical-potential gradients between dendrite and spine head lead to measurable electrical currents. The cable equation, however, considers electrical currents only as result of gradients in the electrical potential. The Poisson-Nernst-Planck (PNP) equations allow a more accurate description, as they include both types of currents. Previous PNP simulations predict a considerable change of ionic concentrations in spines during an excitatory postsynaptic potential (EPSP). However, solving PNP-equations is computationally expensive, limiting their applicability for complex structures. Here, we present a system of equations that generalizes the cable equation and considers both, electrical potentials and time-dependent concentrations of ion species with individual diffusion constants. Still, basic numerical algorithms can be employed to solve such systems. Based on simulations, we confirm that ion concentrations in dendritic spines are changing significantly during current injections that are comparable to synaptic events. Electrical currents reflecting ion diffusion through the spine neck increase voltage depolarizations in the spine head. Based on this effect, we identify a mechanism that affects the influx of Ca2+ in sequences of pre- and postsynaptic action potentials. Taken together, the diffusion of individual ion species need to be taken into account to accurately model electrical currents in dendritic spines. In the future the presented equations can be used to accurately integrate dendritic spines into multicompartment models to study synatptic integration.

Dendritic spines are extremely small and experimentally difficult to access. Therefore, it is still uncertain whether all assumptions of basic neuroscientific theories, such as cable theory, are valid there. Previous theoretical work suggests that electroneutrality could be violated in dendritic spines. If this were true, new theories would be required. Unfortunately, these results were based on a greatly simplified model system with unrealistic ion concentrations. Inspired by these studies, we apply Poison-Nernst-Planck (PNP) equations to study the profiles of ion concentrations and the membrane potential in dendritic spines in a physiologically relevant regime. We find that, for realistic ion concentrations and in contrast to previous results, electroneutrality is a valid assumption for all tested geometries, irrespective of size and shape. However, the surface charge causes an accumulation of counter ions and a strong electric field near the surface of the membrane in the intra- and extracellular space. Still, a plate capacitor model accurately describes the capacitance of the membrane. Most importantly, the two cable parameters - the specific capacitance and the intracellular resistivity - are constants over a wide range of parameters. These results justify the application of models based on cable theory to dendritic spines.