Journal of Computational Neuroscience

In this work, we develop a mathematical model that captures both the early and late phases of Long-Term Potentiation (LTP) and Long-Term Depression (LTD) within an NMDAR-dependent and AMPA-regulated framework. The model combines multiple essential properties. First, it emphasizes a detailed representation of biochemical processes within the postsynaptic neuron, thereby illustrating the interaction between LTD and distinct forms of LTP. Second, the dynamic modulation of postsynaptic AMPA receptor conductance is represented through nonlinear differential equations and algebraic relations. Third, the model incorporates input specificity, associativity, and cooperativity, allowing synaptic changes at one site to influence the strength of neighboring synapses. These features provide a comprehensive description of synaptic dynamics, allowing the simulation of plasticity at both the cellular and the network levels. Overall, the model offers a valuable framework for studying NMDAR-dependent LTP and LTD by explicitly incorporating AMPA currents. We believe that this model provides deeper insights into the molecular mechanisms of synaptic plasticity and paves the way for the construction of network-level models by linking multiple cells through AMPA receptor conductance.

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.

The brain modifies synaptic strengths to store new information via long-term potentiation (LTP) and long-term depression (LTD). Evidence has mounted that long-term plasticity is controlled via concentrations of calcium ([Ca2+]) in postsynaptic spines. Several mathematical models describe this phenomenon, including those of Shouval, Bear, and Cooper (SBC) (Shouval et al., 2002, 2010) and Graupner and Brunel (GB)(Graupner & Brunel, 2012). Here we suggest a generalized version of the SBC and GB models, based on a fixed point - learning rate (FPLR) framework, where the synaptic [Ca2+] specifies a fixed point toward which the synaptic weight approaches asymptotically at a [Ca2+]-dependent rate. The FPLR framework offers a straightforward phenomenological interpretation of calcium-based plasticity: the calcium concentration tells the synaptic weight where it is going and how fast it goes there. The FPLR framework can flexibly incorporate various experimental findings, including the existence of multiple regions of [Ca2+] where no plasticity occurs, or plasticity in cerebellar Purkinje cells, where the directionality of calcium-based synaptic changes is thought to be reversed relative to cortical and hippocampal neurons. We also suggest a modeling approach that captures the dependency of late-phase plasticity stabilization on protein synthesis. We demonstrate that due to the asymptotic, saturating nature of synaptic changes in the FPLR rule, the result of frequency- and spike-timing-dependent plasticity protocols are weight-dependent. Finally, we show how the FPLR framework can explain plateau potential-induced place field formation in hippocampal CA1 neurons, also known as behavioral time scale plasticity (BTSP).

In electrically coupled networks, the coupling coefficient (CC) quantifies the strength of the connectivity or the degree to which two participating nodes are coupled in response to an external input to one of them. The CC is measured by computing the relative responses of the indirectly activated (post-J) and the directly activated (pre-J) nodes. In response to time-dependent inputs, the CC is frequency-dependent and has two components capturing the contributions of the amplitude and phase frequency profiles of the participating nodes (quotient of the amplitudes and phase-difference, respectively). The properties and mechanisms of generation of the frequency-dependent CCs (FD-CCs) are largely unknown beyond electrically coupled passive cells and their electrical circuit equivalents. Being linear and 1D, the FD-CCs for passive cells are relatively simple, consisting of low-pass filters (amplitude) and positive and monotonically increasing phase-difference profiles. In linear systems, the FD-CCs depend on the properties of the pre-J cell and the connectivity and are independent of the properties of the post-J cell and the input amplitude. There is a gap in our understanding of the FD-CCs are shaped by (i) how the presence of intrinsic cellular positive and negative feedback currents and the resulting amplification and resonance phenomena, and (ii) the presence of cellular nonlinearities that incorporates the dependence of the FD-CC on the post-J node in addition to the pre-J one. In this paper we address these issues by using biophysically plausible (conductance-based) mathematical modeling, numerical simulations, analytical calculations and dynamical systems tools. We conduct a systematic analysis of the properties of the FD-CC in networks of two electrically connected nodes receiving oscillatory inputs, which is the minimal network architecture that allows for a systematic study of the biophysical and dynamic mechanisms that shape the FD-CC profiles. The participating neurons are either passive cells (low-pass filters) or resonators (band-pass filter) and exhibit lagging or mixed leading-lagging phase-shift responses as the input frequency increases. The formalism and tools we develop and use in this paper can be extended to larger networks with an arbitrary number of nodes, to spatially extended multicompartment neuronal models, and to neurons having a variety of ionic currents. The principles that emerge from our study are directly applicable to these scenarios. Our results make experimentally testable predictions and have implications for the understanding of spike transmission, synchronized firing and coincidence detection in electrically coupled networks in the presence of oscillatory inputs. For clarity, the paper includes an extensive supplementary material section.

Excitable cells exhibit different electrophysiological profiles while responding to current stimulation in current-clamp experiments. In theory, the differences could be explained by changes in the expression of proteins mediating transmembrane ion transport. Experimental verification by performing systematic, controlled variations in the expression of proteins of the same type (e.g. voltage-dependent, noninactivating Kv2.1 channels) is difficult to achieve in the absence of other changes. However, biophysical models enable this possibility and allows us to assess and characterise the electrophysiological phenotypes associated to different levels of expression of non-inactivating voltage-dependent K-channels of type Kv2.1. To do so, we use a 2-dimensional biophysical model of neuronal membrane potential and study the phase plane geometry and bifurcation structures associated with different levels of Kv2.1 expression with the input current as bifurcation parameter. We find that increasing the expression of Kv2.1 channels reduces the size of the region of the phase plane from which action potentials can be initiated. The changes in expression can also be related to different transitions between rest and repetitive firing in current clamp experiments. For instance, increasing the number of Kv2.1 channels shifts the rheobase current to higher levels, but also expands the dynamic range in which excitatory external current produces repetitive spiking. Our analysis shows that changes in the responses to increasing input currents can be associated to different sequences of fixed point bifurcations. In general, the fixed points are attracting, then repulsive, and later become attracting again as the input current increases, but the bifurcation sequences also include changes in fixed point type, and change qualitatively with the expression of Kv2.1 channels. In the non-repetitive spiking regime with low current stimulation, low expression of Kv2.1 channels yields bifurcation sequences that include transitions between 3 and 1 fixed points, and repetitive firing starts with delays that decrease with increasing current (aggregation). For higher expression of Kv2.1 channels there is only one fixed point that changes in type and attractivity as the input current increases, convergence to rest tends to be oscillatory (resonance), and repetitive spiking starts without noticeable delays. Our models explain how the same neuron is theoretically be capable of including both aggregating and resonant modes of integration for synaptic input, as shown in current clamp experiments.

In the classic view of cortical rhythms, the interaction between excitatory pyramidal neurons (E) and inhibitory parvalbumin neurons (I) has been shown to be sufficient to generate gamma and beta band rhythms. However, it is now clear that there are multiple inhibitory interneuron subtypes and that they play important roles in the generation of these rhythms. In this paper we develop a spiking network that consists of populations of E, I and an additional interneuron type, the somatostatin (S) internerons that receive excitation from the E cells and inhibit both the E cells and the I cells. These S cells are modulated by a third inhibitory subtype, VIP neurons that receive inputs from other cortical areas. We reduce the spiking network to a system of nine differential equations that characterize the mean voltage, firing rate, and synaptic conductance for each population and using this we find many instances of multiple rhythms within the network. Using tools from nonlinear dynamics, we explore the roles of each of the two classes of inhibition as well as the role of the VIP modulation on the properties of these rhythms. Author summaryRhythmic dynamics in the cortex are crucial for information processing, sensory integration, and cognition. In this paper, we look at a model network consisting of a population of excitatory neurons and two distinct populations of inhibitory neurons. We show that the interactions between these three populations gives rise to multiple coexistent rhythms. We also present a greatly simplified model that can be tuned to have similar properties. Our computational model may provide a mechanism for the experimental appearance of multiple rhythms in the same cortical circuit.

Synaptic plasticity involves the modification of both biochemical and structural components of neurons. Many studies have revealed that the change in the number density of the glutamatergic receptor AMPAR at the synapse is proportional to synaptic weight update; increase in AMPAR corresponds to strengthening of synapses while decrease in AMPAR density weakens synaptic connections. The dynamics of AMPAR are thought to be regulated by upstream signaling, primarily the calcium-CaMKII pathway, trafficking to and from the synapse, and influx from extrasynaptic sources. Here, we have developed a set of models using compartmental ordinary differential equations to systematically investigate contributions of signaling and trafficking variations on AMPAR dynamics at the synaptic site. We find that the model properties including network architecture and parameters significantly affect the integration of fast upstream species by slower downstream species. Furthermore, we predict that the model outcome, as determined by bound AMPAR at the synaptic site, depends on (a) the choice of signaling model (bistable CaMKII or monostable CaMKII dynamics), (b) trafficking versus influx contributions, and (c) frequency of stimulus. Therefore, AMPAR dynamics can have unexpected dependencies when upstream signaling dynamics (such as CaMKII and PP1) are coupled with trafficking modalities.

Cartwheel interneurons (CWCs) in the auditory system exhibit a range of activity patterns relevant to auditory function and pathologies. Although experiments have shown how these patterns can vary across individual neurons and can change under pharmacological manipulations, the field has lacked a computational framework in which to explore the contributions of particular currents to these observations and to generate new predictions about the effects of manipulations on CWCs. In this work, we address this deficiency by presenting a conductance-based CWC computational model. This model captures the diversity of CWC activity patterns observed experimentally and suggests parameter changes that may underlie differences across cells. Bifurcation analysis of this model provides an explanation of how distinct dynamic mechanisms contribute to these differences, while direct simulations suggest how cells with different baseline dynamics will respond to variations in certain experimentally-accessible potassium and calcium channel conductances. In addition to the full model that we introduce, we present a reduced model that preserves CWC dynamic regimes. We classify the reduced model variables in terms of distinct dynamic timescales and show that the key transitions in dynamic patterns can be explained based on equilibria of the averaged dynamics of the slowest model variables, in a regime where the faster model variables exhibit oscillations. Overall, this study predicts how changes in parameters will influence CWC behavior, suggests how bifurcations contribute to changes in CWC dynamics, and provides a theoretical foundation that supports our simulation findings. Author summaryCartwheel interneurons (CWCs) are the most common class of inhibitory interneurons in an auditory brainstem region involved in sound localization and are believed to be important for auditory processing and pathologies. Distinct patterns of CWC activity have been observed experimentally in a variety of conditions. In this work, we present two novel computational models that simulate the factors contributing to CWC dynamics. By harnessing this framework, we are able to reveal the contributions of key ion currents to modulating CWC activity. Indeed, we find that the factors present in CWC neurons can produce a complicated dynamic landscape, with a wide range of output patterns possible as the relative strengths of these factors are varied. Overall, our models represent useful tools for understanding experimental results and generating new predictions about CWC behavior. In particular, in the more reduced of the two models, we can perform mathematical analysis to make more detailed predictions about the effects of current modulation on whether CWC neurons will exhibit regular spiking or more complex forms of outputs.

Nonlinear mechanisms are at the heart of neuronal information processing, for example to fire an action potential, the membrane voltage must exceed a threshold nonlinearity. Even though, linear encoding schemes are commonly used and often successfully describe large parts of sensory encoding nonlinear mechanisms such as thresholds and saturations are well known to be crucial to encode behaviorally relevant features in the stimulus space not captured by linear methods. Here we analyze the role of bursts in p-type electroreceptor afferents (P-units) in the weakly electric fish Apteronotus leptorhynchus. It is long known that subpopulations of these cells fire bursts of action potentials while others do not. Previous research suggests, that the non-bursting cells are better at encoding the stimulus time-course while bursting neurons are better suited to encode special features in the stimulus. We here show, based on the analysis of experimental data and modeling, that bursts affect the linear as well as the nonlinear encoding. Theoretical work predicts that in simple leaky-integrate-and-fire model neurons, two periodic stimuli interact nonlinearly when the sum of the two frequencies matches the neurons baseline firing rate as quantified by the second-order susceptibility. Indeed, such nonlinear responses have been found in non-bursting P-units when stimulated by two beats simultaneously but only in those cells, that exhibit very low levels of intrinsic noise. In this study, we found that bursts strongly enhance these nonlinear responses which may play a critical role in the detection of weak intruder signals in the presence of a strong female signal, i.e. an electrosensory cocktail party.

Propagating waves of activity can be evoked and can occur spontaneously in vivo and in vitro. We examine the properties of these waves as inhibition varies in a cortical slice and then develop several computational models. We first show that in the slice, inhibition controls the velocity of propagation as well as the magnitude of the local field potential. We introduce a spiking model of sparsely connected excitatory and inhibitory theta neurons which are distributed on a one-dimensional domain and illustrate both evoked and spontaneous waves. The excitatory neurons have an additional spike-frequency adaptation current which limits their maximal activity. We show that increased inhibition slows the waves down and limits the participation of excitatory cells in this spiking network. Decreased inhibition leads to large amplitude faster moving waves similar to those seen in seizures. To gain further insight into the mechanism that control the waves, we then systematically reduce the model to a Wilson-Cowan type network using a mean-field approach. We simulate this network directly and by using numerical continuation to follow traveling waves in a moving coordinate system as we vary the strength and spread of inhibition and the strength of adaptation. We find several types of instability (bifurcations) that lead to the loss of waves and subsequent pattern formation. We approximate the smooth nonlinearity by a step function and obtain expressions for the velocity, wave-width, and stability. Author summaryStimuli and other aspects of neuronal activity are carried across areas in the brain through the concerted activity of recurrently connected neurons. The activity is controlled through negative feedback from both inhibitory neurons and intrinsic currents in the excitatory neurons. Evoked activity often appears in the form of a traveling pulse of activity. In this paper we study the speed, magnitude, and other properteis of these waves as various aspects of the negative feedback are altered. Inhibition enables information to be readily transmitted across distances without the neural activity blowing up into a seizure-like state.

The subdivision of synaptic vesicles (SVs) into discrete pools is a leading concept of synaptic physiology. To better explain specific properties of transmission and plasticity, it has been suggested initially that the readily releasable pool (RRP) of SVs is subdivided into two parallel pools differing in their release probability. More recently, evidence was provided that sequential pools with a single RRP and a series-connected finite-size replacement pool (RP) inserted between the reserve pool (RSP) and RRP equally well or even better account for most aspects of transmission and plasticity. It was further suggest that a fraction of the presynaptic release sites (N) are initially unoccupied by SVs, with vesicle recruitment occurring rapidly during activity, and furthermore that the number of release sites itself changes with rapid dynamics during activity. Here we propose a framework that identifies specific signs of the presence of the series-connected RP, using a combination of two experimental electrophysiological standard methods, cumulative analysis (CumAna) and multiple probability fluctuation analysis (MPFA). In particular we show that if the y-intercept (y(0)) of CumAna is larger than N reported by MPFA (y(0) > NMPFA) this is a strong indication for a series-connected RP. This is due to the fact that y(0) reports the sum of RRP and RP. Our analysis further suggests that this result is not affected by unoccupied release sites, as such empty sites contribute to both estimates, y(0) and NMPFA. We discuss experimental findings and models in the recent literature in the light of our theoretical considerations.

Paramecium is a large unicellular organism that swims in fresh water using cilia. When stimulated by various means (mechanically, chemically, optically, thermally), it often swims backward then turns and swims forward again in a new direction: this is called the avoiding reaction. This reaction is triggered by a calcium-based action potential. For this reason, several authors have called Paramecium the "swimming neuron". Here we present an empirically constrained model of its action potential based on electrophysiology experiments on live immobilized paramecia, together with simultaneous measurement of ciliary beating using particle image velocimetry. Using these measurements and additional behavioral measurements of free swimming, we extend the electrophysiological model by coupling calcium concentration to kinematic parameters, turning it into a swimming model. In this way, we obtain a model of autonomously behaving Paramecium. Finally, we demonstrate how the modeled organism interacts with an environment, can follow gradients and display collective behavior. This work provides a modeling basis for investigating the physiological basis of autonomous behavior of Paramecium in ecological environments. Author SummaryBehavior depends on a complex interaction between a variety of physiological processes, the body and the environment. We propose to examine this complex interaction in an organism consisting of a single excitable and motile cell, Paramecium. The behavior of Paramecium is based on trial and error: when it encounters an undesirable situation, it backs up and changes direction. This avoiding reaction is triggered by an action potential. Here we developed an empirically constrained biophysical model of Parameciums action potential, which we then coupled to its kinematics. We then demonstrate the potential of this model in investigating various types of autonomous behavior, such as obstacle avoidance, gradient-following and collective behavior.

Purkinje cells are the principal neurons of the cerebellar cortex. One of their distinguishing features is that they fire two distinct types of action potential, called simple and complex spikes, which interact with one another. Simple spikes are stereotypical action potentials that are elicited at high, but variable, rates (0 - 100 Hz) and have a consistent waveform. Complex spikes are composed of an initial action potential followed by a burst of lower amplitude spikelets. Complex spikes occur at comparatively low rates (~ 1 Hz) and have a variable waveform. Although they are critical to cerebellar operation a simple model that describes the complex spike waveform is lacking. Here, a novel single-compartment model of Purkinje cell electrodynamics is presented. The simpler version of this model, with two active conductances and a leak channel, can simulate the features typical of complex spikes recorded in vitro. If calcium dynamics are also included, the model can capture the pause in simple spike activity that occurs after complex spike events. Together, these models provide an insight into the mechanisms behind complex spike spikelet generation, waveform variability and their interactions with simple spike activity.

Voltage perturbations to a repetitively firing Hodgkin-Huxley (HH) model of neuronal spiking in the bistable regime with coexisting limit cycle and stable steady node can either lead to the spikes phase resetting or collapse to the stable steady state. The latter describes a non-firing hyperpolarized quiescent state of the neuron despite the presence of constant external current. Using asymptotic phase response curve (PRC), the impact of voltage perturbations on a repetitively firing HH model is studied here while it is diffusively coupled to another HH model under identical external stimulation. It is observed that the pre-perturbation state of synchronization and the coupling strength critically determine the PRC response of the perturbed HH dynamics. Higher coupling strengths of perfectly in-phase (anti-phase) synchronized HH models shrink (expand) the combinatorial space of perturbation strengths and the oscillation phases causing collapse to the quiescent state. This indicates reduced (enlarged) basin of attraction, viz. the null space, associated with the steady state in the HH phase space. The findings bear important implications to the spiking dynamics of diverse interneurons, as well as special cases of pyramidal neurons, coupled through electrical synapses via. gap junctions, and suggest the role of gap junction plasticity in tuning vulnerability to quiescent state in the presence of biological noise and spikelets.

We formulate a mechanistic model capturing the dynamics of neurotransmitter release in a chemical synapse. The proposed modeling framework captures key aspects such as the random arrival of action potentials (AP) in the presynaptic (input) neuron, probabilistic docking and release of neurotransmitter-filled vesicles, and clearance of the released neurotransmitter from the synaptic cleft. Feedback regulation is implemented by having the released neurotransmitter impact the vesicle docking rate that occurs biologically through "autoreceptors" on the presynaptic membrane. Our analytical results show that these feedbacks can amplify or buffer fluctuations in neurotransmitter levels depending on the relative interplay of neurotransmitter clearance rate with the AP arrival rate and the vesicle replenishment rate, with faster clearance rates leading to noise amplification. We next consider a postsynaptic (output) neuron that fires an AP based on integrating upstream neurotransmitter activity. Investigating the postsynaptic AP firing times, we identify scenarios that lead to band-pass filtering, i.e., the output neuron frequency is maximized at intermediate input neuron frequencies. We extend these results to consider feedforward regulation where in addition to a direct excitatory synapse, the input neuron also impacts the output indirectly via an inhibitory interneuron, and we identify parameter regimes where feedforward neuronal networks result in band-pass filtering.

Computational modeling in neuroscience has largely focused on simulating the electrical activity of neurons, while ignoring other components of brain tissue, such as glial cells and the extracellular space. As such, most existing models can not be used to address pathological conditions, such as spreading depression, which involves dramatic changes in ion concentrations, large extracellular potential gradients, and glial buffering processes. We here present the electrodiffusive neuron-extracellular-glia (edNEG) model, which we believe is the first model to combine multicompartmental neuron modeling with an electrodiffusive framework for intra- and extracellular ion concentration dynamics in a local piece of neuro-glial brain tissue. The edNEG model (i) keeps track of all intraneuronal, intraglial, and extracellular ion concentrations and electrical potentials, (ii) accounts for neuronal somatic action potentials, and dendritic calcium spikes, (iii) contains a neuronal and glial homeostatic machinery that gives physiologically realistic ion concentration dynamics, (iv) accounts for electrodiffusive transmembrane, intracellular, and extracellular ionic movements, and (v) accounts for glial and neuronal swelling caused by osmotic transmembrane pressure gradients. We demonstrate that the edNEG model performs realistically as a local and closed system, i.e., that it maintains a steady state for moderate neural activity, but experiences concentration-dependent effects, such as altered firing patterns and homeostatic breakdown, when the activity level becomes too intense. Furthermore, we study the role of glia in making the neuron more tolerable to hyperactive firing and in limiting neuronal swelling. Finally, we discuss how the edNEG model can be integrated with previous spatial continuum models of spreading depression to account for effects of neuronal morphology, action potential generation, and dendritic Ca2+ spikes which are currently not included in these models. Author summaryNeurons communicate by electrical signals mediated by the movement of ions across the cell membranes. The ionic flow changes the ion concentrations on both sides of the cell membranes, but most modelers of neurons assume ion concentrations to remain constant. Since the neuronal membrane contains structures called ion pumps and cotransporters that work to maintain close-to baseline ion concentrations, and the brain contains a cell type called astrocytes that contribute in keeping an appropriate ionic environment for neurons, the assumption is justifiable in many scenarios. However, for several pathological conditions, such as epilepsy and spreading depression, the ion concentrations may vary dramatically. To study these scenarios, we need models that account for changes in ion concentrations. In this paper, we present what we call the electrodiffusive neuron-extracellular-glia model (edNEG), which keeps track of all ions in a closed system containing a neuron, the extracellular space surrounding it, and an astrocytic "domain". The edNEG model ensures a complete and consistent relationship between ion concentrations and charge conservation. We envision that the model can be used to study a range of pathological conditions such as spreading depression and, hence, be of great value for the field of neuroscience.

In a chemical synapse, information flow occurs via the release of neurotransmitters from a presynaptic neuron that triggers an Action potential (AP) in the postsynaptic neuron. At its core, this occurs via the postsynaptic membrane potential integrating neurotransmitter-induced synaptic currents, and AP generation occurs when potential reaches a critical threshold. This manuscript investigates feedback implementation via an autapse, where the axon from the postsynaptic neuron forms an inhibitory synapse onto itself. Using a stochastic model of neuronal synaptic transmission, we formulate AP generation as a first-passage time problem and derive expressions for both the mean and noise of AP-firing times. Our analytical results supported by stochastic simulations identify parameter regimes where autaptic feedback transmission enhances the precision of AP firing times consistent with experimental data. These noise attenuating regimes are intuitively based on two orthogonal mechanisms - either expanding the time window to integrate noisy upstream signals; or by linearizing the mean voltage increase over time. Interestingly, we find regimes for noise amplification that specifically occur when the inhibitory synapse has a low probability of release for synaptic vesicles. In summary, this work explores feedback modulation of the stochastic dynamics of autaptic neurotransmission and reveals its function of creating more regular AP firing patterns.

Understanding how neuronal networks process oscillatory inputs is key for deciphering the brains information processing dynamics. Neuronal filters describe the frequency-dependent relationship of neuronal outputs (e.g., membrane potential amplitude, firing rate) and their inputs for the level of neuronal organization (e.g., cellular, network) considered. Band-pass filters are associated to the notion of resonance and reflect the systems ability to respond maximally to inputs at a nonzero (resonant) frequency or a limited (resonant) frequency band. The complementary notion of phasonance refers to the ability of a system to exhibit a zero-phase response for a nonzero (phasonant) input frequency. The biophysical and dynamic mechanisms that shape neuronal filters and give raise to preferred frequency responses to oscillatory inputs are poorly understood beyond single cells. Moreover, the mechanisms that control the frequency-dependent communication of information across cells in a network remain unclear. Here, we use mathematical modeling, analytical calculations, computational simulations and dynamical systems tools to investigate how the complex and nonlinear interaction of the systemss biophysical properties and interacting time scales shape neuronal filters in minimal network models receiving oscillatory inputs with frequencies (f) within some range. The minimal networks consist of one directly stimulated cell (cell 1) connected to another (not directly stimulated) cell (cell 2) via graded chemical synapses. Individual cells are either passive or resonators and chemical synapses are either excitatory or inhibitory. The network outputs consist of the voltage peak envelopes and the impedance amplitude and phase profiles (as a function of f) for the two cells. We introduce the frequency-dependent amplitude K(f) and phase {Delta}{Phi}(f) communication coefficients, defined as the ratio of the amplitude responses of the indirectly and directly stimulated cells and the phase difference between these two cells, respectively. Extending previous work, we also introduce the K-curve, parametrized by f, in the phase-space diagram for the voltage variables of the two participating cells. This curve joins the peak voltage values of the two cells in response to the oscillatory inputs and is a geometric representation of the communication coefficient. It allows to interpret the results and explain the dependence of the properties of the communication coefficient in terms of the biophysical and dynamic properties of the participating cells and synaptic connectivity when analytical calculations are not possible. We describe the conditions under which one or the two cells in the network exhibit resonance and phasonance and the conditions under which the network exhibits K-resonance and {Delta}{Phi}-phasonance and more complex network responses depending as the complexity of the participating cells increases. For linear networks (linear nodes and linear connectivity), K is proportional to the impedance of the indirectly activated cell 2 and {Delta}{Phi} is equal to the phase of the indirectly stimulated cell 2, independent of the directly stimulated cell 1 in both cases. We show that the presence of nonlinear connectivity in the network creates (nonlinear) interactions between the two cells that give rise to K-resonance, {Delta}{Phi}-phasonance and more complex responses that are absent in the corresponding linear networks. The results and methods developed in this paper have implications for the processing of information in more complex networks.

AO_SCPLOWBSTRACTC_SCPLOWBoth subtractive and divisive inhibition has been recorded in cortical circuits and recent findings suggest that different interneuronal populations are responsible for the different types of inhibition. This calls for the formulation and description of these inhibitory mechanisms in computational models of cortical networks. Neural mass and neural field models typically only feature subtractive inhibition. Here, we introduce how divisive inhibition can be incorporated in such models, using the Wilson-Cowan modelling formalism as an example. In addition, we show how the subtractive and divisive modulations can be combined. Including divisive inhibition in neural mass models is a crucial step in understanding its role in shaping oscillatory phenomena in cortical networks.

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.