Alexey Kuznetsov

Exploring Neuronal Bistability at the Depolarization Block

Many neurons display bistability–coexistence of two firing modes such as bursting and tonic spiking or tonic spiking and silence. Bistability has been proposed to endow neurons with richer forms of information processing in general and to be involved in short-term memory in particular by allowing a brief signal to elicit long-lasting changes in firing. In this paper, we focus on bistability that allows for a choice between tonic spiking and depolarization block in a wide range of the depolarization levels. We consider the spike-producing currents in two neurons, models of which differ by the parameter values. Our dopaminergic neuron model displays bistability in a wide range of applied currents at the depolarization block. The Hodgkin-Huxley model of the squid giant axon shows no bistability. We varied parameter values for the model to analyze transitions between the two parameter sets. We show that bistability primarily characterizes the inactivation of the Na+ current. Our study suggests a connection between the amount of the Na+ window current and the length of the bistability range. For the dopaminergic neuron we hypothesize that bistability can be linked to a prolonged action of antipsychotic drugs.

Interaction of NMDA Receptor and Pacemaking Mechanisms in the Midbrain Dopaminergic Neuron

Dopamine neurotransmission has been found to play a role in addictive behavior and is altered in psychiatric disorders. Dopaminergic (DA) neurons display two functionally distinct modes of electrophysiological activity: low- and high-frequency firing. A puzzling feature of the DA neuron is the following combination of its responses: N-methyl-D-aspartate receptor (NMDAR) activation evokes high-frequency firing, whereas other tonic excitatory stimuli (-amino-3-hydroxyl-5-methyl-4-isoxazolepropionate receptor (AMPAR) activation or applied depolarization) block firing instead. We suggest a new computational model that reproduces this combination of responses and explains recent experimental data. Namely, somatic NMDAR stimulation evokes high-frequency firing and is more effective than distal dendritic stimulation. We further reduce the model to a single compartment and analyze the mechanism of the distinct high-frequency response to NMDAR activation vs. other stimuli. Standard nullcline analysis shows that the mechanism is based on a decrease in the amplitude of calcium oscillations. The analysis confirms that the nonlinear voltage dependence provided by the magnesium block of the NMDAR determine its capacity to elevate the firing frequency. We further predict that the moderate slope of the voltage dependence plays the central role in the frequency elevation. Additionally, we suggest a repolarizing current that sustains calcium-independent firing or firing in the absence of calcium-dependent repolarizing currents. We predict that the ether–a-go-go current (ERG), which has been observed in the DA neuron, is the best fit for this critical role. We show that a calcium-dependent and a calcium-independent oscillatory mechanisms form a structure of interlocked negative feedback loops in the DA neuron. The structure connects research of DA neuron firing with circadian biology and determines common minimal models for investigation of robustness of oscillations, which is critical for normal function of both systems.

Dopamine Neurons Change the Type of Excitability in Response to Stimuli

The dynamics of neuronal excitability determine the neuron’s response to stimuli, its synchronization and resonance properties and, ultimately, the computations it performs in the brain. We investigated the dynamical mechanisms underlying the excitability type of dopamine (DA) neurons, using a conductance-based biophysical model, and its regulation by intrinsic and synaptic currents. Calibrating the model to reproduce low frequency tonic firing results in N-methyl-D-aspartate (NMDA) excitation balanced by γ-Aminobutyric acid (GABA)-mediated inhibition and leads to type I excitable behavior characterized by a continuous decrease in firing frequency in response to hyperpolarizing currents. Furthermore, we analyzed how excitability type of the DA neuron model is influenced by changes in the intrinsic current composition. A subthreshold sodium current is necessary for a continuous frequency decrease during application of a negative current, and the low-frequency “balanced” state during simultaneous activation of NMDA and GABA receptors. Blocking this current switches the neuron to type II characterized by the abrupt onset of repetitive firing. Enhancing the anomalous rectifier Ih current also switches the excitability to type II. Key characteristics of synaptic conductances that may be observed in vivo also change the type of excitability: a depolarized γ-Aminobutyric acid receptor (GABAR) reversal potential or co-activation of α-amino-3-hydroxy-5-methyl-4-isoxazolepropionic acid receptors (AMPARs) leads to an abrupt frequency drop to zero, which is typical for type II excitability. Coactivation of N-methyl-D-aspartate receptors (NMDARs) together with AMPARs and GABARs shifts the type I/II boundary toward more hyperpolarized GABAR reversal potentials. To better understand how altering each of the aforementioned currents leads to changes in excitability profile of DA neuron, we provide a thorough dynamical analysis. Collectively, these results imply that type I excitability in dopamine neurons might be important for low firing rates and fine-tuning basal dopamine levels, while switching excitability to type II during NMDAR and AMPAR activation may facilitate a transient increase in dopamine concentration, as type II neurons are more amenable to synchronization by mutual excitation.

An interlocked oscillator model for firing of the mesencephalic dopaminergic neuron.

Dopaminergic (DA) neurons display two functionally distinct modes of electrical activity: low- and high-frequency firing. We suggest a new minimal computational model that unites data on these firing modes obtained under different experimental conditions. The model reproduces the separation of maximal frequencies under NMDA synaptic stimulation vs. other treatments. In accord to recent experimental data, NMDA stimulation restricted to the soma effectively evokes high-frequency oscillations in the model. We have also reproduced low- and high-frequency oscillations under blockade of the SK current. Thus, the new model suggests a way that overcomes all major limitations of the switching dominance mechanism for controlling the frequency of the DA neuron. We explain recent experimental facts and make further predictions.

Synergy of AMPA and NMDA Receptor Currents in Dopaminergic Neurons: A Modeling Study.

Dopaminergic (DA) neurons display two modes of firing: low-frequency tonic and high-frequency bursts. The high frequency firing within the bursts is attributed to NMDA, but not AMPA receptor activation. In our models of the DA neuron, both biophysical and abstract, the NMDA receptor current can significantly increase their firing frequency, whereas the AMPA receptor current is not able to evoke high-frequency activity and usually suppresses firing. However, both currents are produced by glutamate receptors and, consequently, are often co-activated. Here we consider combined influence of AMPA and NMDA synaptic input in the models of the DA neuron. Different types of neuronal activity (resting state, low frequency, or high frequency firing) are observed depending on the conductance of the AMPAR and NMDAR currents. In two models, biophysical and reduced, we show that the firing frequency increases more effectively if both receptors are co-activated for certain parameter values. In particular, in the more quantitative biophysical model, the maximal frequency is 40% greater than that with NMDAR alone. The dynamical mechanism of such frequency growth is explained in the framework of phase space evolution using the reduced model. In short, both the AMPAR and NMDAR currents flatten the voltage nullcline, providing the frequency increase, whereas only NMDA prevents complete unfolding of the nullcline, providing robust firing. Thus, we confirm a major role of the NMDAR in generating high-frequency firing and conclude that AMPAR activation further significantly increases the frequency.

A minimal model for a slow pacemaking neuron

Abstract We have constructed a phenomenological model for slow pacemaking neurons. These are neurons that generate very regular periodic oscillations of the membrane potential. Many of these neurons also differentially respond to various types of stimulation. The model is based on FitzHugh–Nagumo (FHN) oscillator and implements a nonlinearity introduced by a current that depends on an ion concentration. The comparison with the original FHN oscillator has shown that the new nonlinear dependence allows for differentiating responses to various stimuli. We discuss implications of our results for a broad class of neurons.

Frequency switching in a two-compartmental model of the dopaminergic neuron

Mid-brain dopaminergic (DA) neurons display two functionally distinct modes of electrical activity: low- and high-frequency firing. The high-frequency firing is linked to important behavioral events in vivo. However, it cannot be elicited by standard manipulations in vitro. We had suggested a two-compartmental model of the DA cell that united data on firing frequencies under different experimental conditions. We now analyze dynamics of this model. The analysis was possible due to introduction of timescale separation among variables. We formulate the requirements for low and high frequencies. We found that the modulation of the SK current gating controls the frequency rise under applied depolarization. This provides a new mechanism that limits the frequency in the control conditions and allows high-frequency responses to depolarization if the SK current gating is downregulated. The mechanism is based on changing Ca2u2009+u2009 balance and can also be achieved by direct modulation of the balance. Interestingly, such changes do not affect the high-frequency oscillations under NMDA. Therefore, altering Ca2u2009+u2009 balance allows combining the high-frequency response to NMDA activation with the inability of other treatments to effectively elevate the frequency. We conclude that manipulations affecting Ca2u2009+u2009 balance are most effective in controlling the frequency range. This modeling prediction gives a clue to the mechanism of the high-frequency firing in the DA neuron in vivo and in vitro.

Implications of cellular models of dopamine neurons for disease

This review addresses the present state of single-cell models of the firing pattern of midbrain dopamine neurons and the insights that can be gained from these models into the underlying mechanisms for diseases such as Parkinsons, addiction, and schizophrenia. We will explain the analytical technique of separation of time scales and show how it can produce insights into mechanisms using simplified single-compartment models. We also use morphologically realistic multicompartmental models to address spatially heterogeneous aspects of neural signaling and neural metabolism. Separation of time scale analyses are applied to pacemaking, bursting, and depolarization block in dopamine neurons. Differences in subpopulations with respect to metabolic load are addressed using multicompartmental models.

Chaos and robustness in a single family of genetic oscillatory networks.

Genetic oscillatory networks can be mathematically modeled with delay differential equations (DDEs). Interpreting genetic networks with DDEs gives a more intuitive understanding from a biological standpoint. However, it presents a problem mathematically, for DDEs are by construction infinitely-dimensional and thus cannot be analyzed using methods common for systems of ordinary differential equations (ODEs). In our study, we address this problem by developing a method for reducing infinitely-dimensional DDEs to two- and three-dimensional systems of ODEs. We find that the three-dimensional reductions provide qualitative improvements over the two-dimensional reductions. We find that the reducibility of a DDE corresponds to its robustness. For non-robust DDEs that exhibit high-dimensional dynamics, we calculate analytic dimension lines to predict the dependence of the DDEs’ correlation dimension on parameters. From these lines, we deduce that the correlation dimension of non-robust DDEs grows linearly with the delay. On the other hand, for robust DDEs, we find that the period of oscillation grows linearly with delay. We find that DDEs with exclusively negative feedback are robust, whereas DDEs with feedback that changes its sign are not robust. We find that non-saturable degradation damps oscillations and narrows the range of parameter values for which oscillations exist. Finally, we deduce that natural genetic oscillators with highly-regular periods likely have solely negative feedback.

Dynamical ventral tegmental area circuit mechanisms of alcohol-dependent dopamine release

A large body of data has identified numerous molecular targets through which ethanol (EtOH) acts on brain circuits. Yet how these multiple mechanisms interact to result in dysregulated dopamine (DA) release under the influence of alcohol in vivo remains unclear. In this manuscript, we delineate potential circuit‐level mechanisms responsible for EtOH‐dependent dysregulation of DA release from the ventral tegmental area (VTA) into its projection areas. For this purpose, we constructed a circuit model of the VTA that integrates realistic Glutamatergic (Glu) inputs and reproduces DA release observed experimentally. We modelled the concentration‐dependent effects of EtOH on its principal VTA targets. We calibrated the model to reproduce the inverted U‐shape dose dependence of DA neuron activity on EtOH concentration. The model suggests a primary role of EtOH‐induced boost in the Ih and AMPA currents in the DA firing‐rate/bursting increase. This is counteracted by potentiated GABA transmission that decreases DA neuron activity at higher EtOH concentrations. Thus, the model connects well‐established in vitro pharmacological EtOH targets with its in vivo influence on neuronal activity. Furthermore, we predict that increases in VTA activity produced by moderate EtOH doses require partial synchrony and relatively low rates of the Glu afferents. We propose that the increased frequency of transient (phasic) DA peaks evoked by EtOH results from synchronous population bursts in VTA DA neurons. Our model predicts that the impact of acute ETOH on dopamine release is critically shaped by the structure of the cortical inputs to the VTA.