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Abstract Sodium ions (Na⁺) are major charge carriers mediating neuronal excitation and play a fundamental role in brain physiology. Glutamatergic synaptic activity is accompanied by large transient Na⁺increases, but the spatio-temporal dynamics of Na⁺signals and properties of Na⁺diffusion within dendrites are largely unknown. To address these questions, we employed multi-photon Na⁺imaging combined with whole-cell patch-clamp in dendrites of CA1 pyramidal neurons in tissue slices from mice of both sexes. Fluorescence lifetime microscopy revealed a dendritic baseline Na⁺concentration of ~10 mM. Using intensity-based line-scan imaging we found that local, glutamate-evoked Na⁺signals spread rapidly within dendrites, with peak amplitudes decreasing and latencies increasing with increasing distance from the site of stimulation. Spread of Na⁺along dendrites was independent of dendrite diameter, order or overall spine density in the ranges measured. Our experiments also show that dendritic Na⁺readily invades spines and suggest that spine necks may represent a partial diffusion barrier. Experimental data were well reproduced by mathematical simulations assuming normal diffusion with a diffusion coefficient of. Modeling moreover revealed that lateral diffusion is key for the clearance of local Na⁺increases at early time points, whereas when diffusional gradients are diminished, Na⁺/K⁺-ATPase becomes more relevant. Taken together, our study thus demonstrates that Na⁺influx causes rapid lateral diffusion of Na⁺within spiny dendrites. This results in an efficient redistribution and fast recovery from local Na⁺transients which is mainly governed by concentration differences. Significance statementActivity of excitatory glutamatergic synapses generates large Na⁺transients in postsynaptic cells. Na⁺influx is a main driver of energy consumption and modulates cellular properties by modulating Na⁺-dependent transporters. Knowing the spatio-temporal dynamics of dendritic Na⁺signals is thus critical for understanding neuronal function. To study propagation of Na⁺signals within spiny dendrites, we performed fast Na⁺imaging combined with mathematical simulations. Our data shows that normal diffusion, based on a diffusion coefficient of 600 µm²/s, is crucial for fast clearance of local Na⁺transients in dendrites, whereas Na⁺export by the Na⁺/K⁺-ATPase becomes more relevant at later time points. This fast diffusive spread of Na⁺will reduce the local metabolic burden imposed by synaptic Na⁺influx. 

There is an increasing need to implement neuromorphic systems that are both energetically and computationally efficient. There is also great interest in using electric elements with memory, memelements, that can implement complex neuronal functions intrinsically. A feature not widely incorporated in neuromorphic systems is history-dependent action potential time adaptation which is widely seen in real cells. Previous theoretical work shows that power-law history dependent spike time adaptation, seen in several brain areas and species, can be modeled with fractional order differential equations. Here, we show that fractional order spiking neurons can be implemented using super-capacitors. The super-capacitors have fractional order derivative and memcapacitive properties. We implemented two circuits, a leaky integrate and fire and a Hodgkin–Huxley. Both circuits show power-law spiking time adaptation and optimal coding properties. The spiking dynamics reproduced previously published computer simulations. However, the fractional order Hodgkin–Huxley circuit showed novel dynamics consistent with criticality. We compared the responses of this circuit to recordings from neurons in the weakly-electric fish that have previously been shown to perform fractional order differentiation of their sensory input. The criticality seen in the circuit was confirmed in spontaneous recordings in the live fish. Furthermore, the circuit also predicted long-lasting stimulation that was also corroborated experimentally. Our work shows that fractional order memcapacitors provide intrinsic memory dependence that could allow implementation of computationally efficient neuromorphic devices. Memcapacitors are static elements that consume less energy than the most widely studied memristors, thus allowing the realization of energetically efficient neuromorphic devices. 

In recent years, the field of neuroscience has gone through rapid experimental advances and a significant increase in the use of quantitative and computational methods. This growth has created a need for clearer analyses of the theory and modeling approaches used in the field. This issue is particularly complex in neuroscience because the field studies phenomena that cross a wide range of scales and often require consideration at varying degrees of abstraction, from precise biophysical interactions to the computations they implement. We argue that a pragmatic perspective of science, in which descriptive, mechanistic, and normative models and theories each play a distinct role in defining and bridging levels of abstraction, will facilitate neuroscientific practice. This analysis leads to methodological suggestions, including selecting a level of abstraction that is appropriate for a given problem, identifying transfer functions to connect models and data, and the use of models themselves as a form of experiment.