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1.  Homeostatic reinforcement learning for integrating reward collection and physiological stability 
eLife  null;3:e04811.
Efficient regulation of internal homeostasis and defending it against perturbations requires adaptive behavioral strategies. However, the computational principles mediating the interaction between homeostatic and associative learning processes remain undefined. Here we use a definition of primary rewards, as outcomes fulfilling physiological needs, to build a normative theory showing how learning motivated behaviors may be modulated by internal states. Within this framework, we mathematically prove that seeking rewards is equivalent to the fundamental objective of physiological stability, defining the notion of physiological rationality of behavior. We further suggest a formal basis for temporal discounting of rewards by showing that discounting motivates animals to follow the shortest path in the space of physiological variables toward the desired setpoint. We also explain how animals learn to act predictively to preclude prospective homeostatic challenges, and several other behavioral patterns. Finally, we suggest a computational role for interaction between hypothalamus and the brain reward system.
eLife digest
Our survival depends on our ability to maintain internal states, such as body temperature and blood sugar levels, within narrowly defined ranges, despite being subject to constantly changing external forces. This process, which is known as homeostasis, requires humans and other animals to carry out specific behaviors—such as seeking out warmth or food—to compensate for changes in their environment. Animals must also learn to prevent the potential impact of changes that can be anticipated.
A network that includes different regions of the brain allows animals to perform the behaviors that are needed to maintain homeostasis. However, this network is distinct from the network that supports the learning of new behaviors in general. These two systems must, therefore, interact so that animals can learn novel strategies to support their physiological stability, but it is not clear how animals do this.
Keramati and Gutkin have now devised a mathematical model that explains the nature of this interaction, and that can account for many behaviors seen among animals, even those that might otherwise appear irrational. There are two assumptions at the heart of the model. First, it is assumed that animals are capable of guessing the impact of the outcome of their behaviors on their internal state. Second, it is assumed that animals find a behavior rewarding if they believe that the predicted impact of its outcome will reduce the difference between a particular internal state and its ideal value. For example, a form of behavior for a human might be going to the kitchen, and an outcome might be eating chocolate.
Based on these two assumptions, the model shows that animals stabilize their internal state around its ideal value by simply learning to perform behaviors that lead to rewarding outcomes (such as going into the kitchen and eating chocolate). Their theory also explains the physiological importance of a type of behavior known as ‘delay discounting’. Animals displaying this form of behavior regard a positive outcome as less rewarding the longer they have to wait for it. The model proves mathematically that delay discounting is a logical way to optimize homeostasis.
In addition to making a number of predictions that could be tested in experiments, Keramati and Gutkin argue that their model can account for the failure of homeostasis to limit food consumption whenever foods loaded with salt, sugar or fat are freely available.
PMCID: PMC4270100  PMID: 25457346
reinforcement learning; homeostatic regulation; cortico-basal ganglia; hypothalamus; temporal discounting; anticipatory responding; None
2.  Endogenous Cholinergic Inputs and Local Circuit Mechanisms Govern the Phasic Mesolimbic Dopamine Response to Nicotine 
PLoS Computational Biology  2013;9(8):e1003183.
Nicotine exerts its reinforcing action by stimulating nicotinic acetylcholine receptors (nAChRs) and boosting dopamine (DA) output from the ventral tegmental area (VTA). Recent data have led to a debate about the principal pathway of nicotine action: direct stimulation of the DAergic cells through nAChR activation, or disinhibition mediated through desensitization of nAChRs on GABAergic interneurons. We use a computational model of the VTA circuitry and nAChR function to shed light on this issue. Our model illustrates that the α4β2-containing nAChRs either on DA or GABA cells can mediate the acute effects of nicotine. We account for in vitro as well as in vivo data, and predict the conditions necessary for either direct stimulation or disinhibition to be at the origin of DA activity increases. We propose key experiments to disentangle the contribution of both mechanisms. We show that the rate of endogenous acetylcholine input crucially determines the evoked DA response for both mechanisms. Together our results delineate the mechanisms by which the VTA mediates the acute rewarding properties of nicotine and suggest an acetylcholine dependence hypothesis for nicotine reinforcement.
Author Summary
Nicotine is the major addictive substance in tobacco smoke. Nicotine exerts its control over neural circuits through nicotinic acetylcholine receptors that normally respond to endogenous acetylcholine. Activation of dopamine neurons in the mesolimbic dopaminergic circuits, which signal motivational properties of actions and stimuli, is at the heart of mediating nicotine reward and dependence. However, major questions have remained unsettled over the precise mechanisms by which nicotine usurps dopaminergic signaling: through receptor activation on dopamine neurons or through receptor desensitization on local inhibitory interneurons. Here we reconcile this debate by showing that both mechanisms are possible. Most notably we present a novel hypothesis suggesting that the mechanisms for nicotine action are state-dependent; they are controlled by the rate of the endogenous cholinergic input to the dopaminergic circuits.
PMCID: PMC3744411  PMID: 23966848
3.  Analytical Insights on Theta-Gamma Coupled Neural Oscillators 
In this paper, we study the dynamics of a quadratic integrate-and-fire neuron, spiking in the gamma (30–100 Hz) range, coupled to a delta/theta frequency (1–8 Hz) neural oscillator. Using analytical and semianalytical methods, we were able to derive characteristic spiking times for the system in two distinct regimes (depending on parameter values): one regime where the gamma neuron is intrinsically oscillating in the absence of theta input, and a second one in which gamma spiking is directly gated by theta input, i.e., windows of gamma activity alternate with silence periods depending on the underlying theta phase. In the former case, we transform the equations such that the system becomes analogous to the Mathieu differential equation. By solving this equation, we can compute numerically the time to the first gamma spike, and then use singular perturbation theory to find successive spike times. On the other hand, in the excitable condition, we make direct use of singular perturbation theory to obtain an approximation of the time to first gamma spike, and then extend the result to calculate ensuing gamma spikes in a recursive fashion. We thereby give explicit formulas for the onset and offset of gamma spike burst during a theta cycle, and provide an estimation of the total number of spikes per theta cycle both for excitable and oscillator regimes.
PMCID: PMC3848946  PMID: 23945442
Oscillations; PING; Dynamical systems; Geometric singular perturbation theory; Blow-up method; Spike times; Theta-gamma rhythms; Type I neuron; SNIC bifurcation
6.  Imbalanced Decision Hierarchy in Addicts Emerging from Drug-Hijacked Dopamine Spiraling Circuit 
PLoS ONE  2013;8(4):e61489.
Despite explicitly wanting to quit, long-term addicts find themselves powerless to resist drugs, despite knowing that drug-taking may be a harmful course of action. Such inconsistency between the explicit knowledge of negative consequences and the compulsive behavioral patterns represents a cognitive/behavioral conflict that is a central characteristic of addiction. Neurobiologically, differential cue-induced activity in distinct striatal subregions, as well as the dopamine connectivity spiraling from ventral striatal regions to the dorsal regions, play critical roles in compulsive drug seeking. However, the functional mechanism that integrates these neuropharmacological observations with the above-mentioned cognitive/behavioral conflict is unknown. Here we provide a formal computational explanation for the drug-induced cognitive inconsistency that is apparent in the addicts' “self-described mistake”. We show that addictive drugs gradually produce a motivational bias toward drug-seeking at low-level habitual decision processes, despite the low abstract cognitive valuation of this behavior. This pathology emerges within the hierarchical reinforcement learning framework when chronic exposure to the drug pharmacologically produces pathologicaly persistent phasic dopamine signals. Thereby the drug hijacks the dopaminergic spirals that cascade the reinforcement signals down the ventro-dorsal cortico-striatal hierarchy. Neurobiologically, our theory accounts for rapid development of drug cue-elicited dopamine efflux in the ventral striatum and a delayed response in the dorsal striatum. Our theory also shows how this response pattern depends critically on the dopamine spiraling circuitry. Behaviorally, our framework explains gradual insensitivity of drug-seeking to drug-associated punishments, the blocking phenomenon for drug outcomes, and the persistent preference for drugs over natural rewards by addicts. The model suggests testable predictions and beyond that, sets the stage for a view of addiction as a pathology of hierarchical decision-making processes. This view is complementary to the traditional interpretation of addiction as interaction between habitual and goal-directed decision systems.
PMCID: PMC3634778  PMID: 23637842
7.  Impact of prefrontal cortex in nicotine-induced excitation of VTA dopamine neurons in anesthetized rats 
Systemic administration of nicotine increases dopaminergic (DA) neuron firing in the ventral tegmental area (VTA), which is thought to underlie nicotine reward. Here, we report that the medial prefrontal cortex (mPFC) plays a critical role in nicotine-induced excitation of VTA DA neurons. In chloral hydrate-anesthetized rats, extracellular single-unit recordings showed that VTA DA neurons exhibited two types of firing responses to systemic nicotine. After nicotine injection, the neurons with type-I response showed a biphasic early inhibition and later excitation, whereas the neurons with type-II response showed a monophasic excitation. The neurons with type-I, but not type-II, response exhibited pronounced slow oscillations (SO) in firing. Pharmacological or structural mPFC inactivation abolished SO and prevented systemic nicotine-induced excitation in the neurons with type-I, but not type-II, response, suggesting that these VTA DA neurons are functionally coupled to the mPFC and nicotine increases firing rate in these neurons in part through the mPFC. Systemic nicotine also increased the firing rate and SO in mPFC pyramidal neurons. mPFC infusion of a non-α7 nAChR antagonist mecamylamine blocked the excitatory effect of systemic nicotine on the VTA DA neurons with type-I response, but mPFC infusion of nicotine failed to excite these neurons. These results suggest that nAChR activation in the mPFC is necessary, but not sufficient, for systemic nicotine-induced excitation of VTA neurons. Finally, systemic injection of bicuculline prevented nicotine-induced firing alterations in the neurons with type-I response. We propose that the mPFC plays a critical role in systemic nicotine-induced excitation of VTA DA neurons.
PMCID: PMC3516404  PMID: 22956827
nicotine; prefrontal cortex; ventral tegmental area; dopamine neuron; in vivo recording; slow oscillation
8.  Passive Dendrites Enable Single Neurons to Compute Linearly Non-separable Functions 
PLoS Computational Biology  2013;9(2):e1002867.
Local supra-linear summation of excitatory inputs occurring in pyramidal cell dendrites, the so-called dendritic spikes, results in independent spiking dendritic sub-units, which turn pyramidal neurons into two-layer neural networks capable of computing linearly non-separable functions, such as the exclusive OR. Other neuron classes, such as interneurons, may possess only a few independent dendritic sub-units, or only passive dendrites where input summation is purely sub-linear, and where dendritic sub-units are only saturating. To determine if such neurons can also compute linearly non-separable functions, we enumerate, for a given parameter range, the Boolean functions implementable by a binary neuron model with a linear sub-unit and either a single spiking or a saturating dendritic sub-unit. We then analytically generalize these numerical results to an arbitrary number of non-linear sub-units. First, we show that a single non-linear dendritic sub-unit, in addition to the somatic non-linearity, is sufficient to compute linearly non-separable functions. Second, we analytically prove that, with a sufficient number of saturating dendritic sub-units, a neuron can compute all functions computable with purely excitatory inputs. Third, we show that these linearly non-separable functions can be implemented with at least two strategies: one where a dendritic sub-unit is sufficient to trigger a somatic spike; another where somatic spiking requires the cooperation of multiple dendritic sub-units. We formally prove that implementing the latter architecture is possible with both types of dendritic sub-units whereas the former is only possible with spiking dendrites. Finally, we show how linearly non-separable functions can be computed by a generic two-compartment biophysical model and a realistic neuron model of the cerebellar stellate cell interneuron. Taken together our results demonstrate that passive dendrites are sufficient to enable neurons to compute linearly non-separable functions.
Author Summary
Classical views on single neuron computation treat dendrites as mere collectors of inputs, that is forwarded to the soma for linear summation and causes a spike output if it is sufficiently large. Such a single neuron model can only compute linearly separable input-output functions, representing a small fraction of all possible functions. Recent experimental findings show that in certain pyramidal cells excitatory inputs can be supra-linearly integrated within a dendritic branch, turning this branch into a spiking dendritic sub-unit. Neurons containing many of these dendritic sub-units can compute both linearly separable and linearly non-separable functions. Nevertheless, other neuron types have dendrites which do not spike because the required voltage gated channels are absent. However, these dendrites sub-linearly sum excitatory inputs turning branches into saturating sub-units. We wanted to test if this last type of non-linear summation is sufficient for a single neuron to compute linearly non-separable functions. Using a combination of Boolean algebra and biophysical modeling, we show that a neuron with a single non-linear dendritic sub-unit whether spiking or saturating is able to compute linearly non-separable functions. Thus, in principle, any neuron with a dendritic tree, even passive, can compute linearly non-separable functions.
PMCID: PMC3585427  PMID: 23468600
9.  Correlations in background activity control persistent state stability and allow execution of working memory tasks 
Working memory (WM) requires selective information gating, active information maintenance, and rapid active updating. Hence performing a WM task needs rapid and controlled transitions between neural persistent activity and the resting state. We propose that changes in correlations in neural activity provides a mechanism for the required WM operations. As a proof of principle, we implement sustained activity and WM in recurrently coupled spiking networks with neurons receiving excitatory random background activity where background correlations are induced by a common noise source. We first characterize how the level of background correlations controls the stability of the persistent state. With sufficiently high correlations, the sustained state becomes practically unstable, so it cannot be initiated by a transient stimulus. We exploit this in WM models implementing the delay match to sample task by modulating flexibly in time the correlation level at different phases of the task. The modulation sets the network in different working regimes: more prompt to gate in a signal or clear the memory. We examine how the correlations affect the ability of the network to perform the task when distractors are present. We show that in a winner-take-all version of the model, where two populations cross-inhibit, correlations make the distractor blocking robust. In a version of the mode where no cross inhibition is present, we show that appropriate modulation of correlation levels is sufficient to also block the distractor access while leaving the relevant memory trace in tact. The findings presented in this manuscript can form the basis for a new paradigm about how correlations are flexibly controlled by the cortical circuits to execute WM operations.
PMCID: PMC3801087  PMID: 24155714
correlations; background activity; working memory; spiking neural network; persistent activity
10.  Spike-Timing Dependent Plasticity and Feed-Forward Input Oscillations Produce Precise and Invariant Spike Phase-Locking 
In the hippocampus and the neocortex, the coupling between local field potential (LFP) oscillations and the spiking of single neurons can be highly precise, across neuronal populations and cell types. Spike phase (i.e., the spike time with respect to a reference oscillation) is known to carry reliable information, both with phase-locking behavior and with more complex phase relationships, such as phase precession. How this precision is achieved by neuronal populations, whose membrane properties and total input may be quite heterogeneous, is nevertheless unknown. In this note, we investigate a simple mechanism for learning precise LFP-to-spike coupling in feed-forward networks – the reliable, periodic modulation of presynaptic firing rates during oscillations, coupled with spike-timing dependent plasticity. When oscillations are within the biological range (2–150 Hz), firing rates of the inputs change on a timescale highly relevant to spike-timing dependent plasticity (STDP). Through analytic and computational methods, we find points of stable phase-locking for a neuron with plastic input synapses. These points correspond to precise phase-locking behavior in the feed-forward network. The location of these points depends on the oscillation frequency of the inputs, the STDP time constants, and the balance of potentiation and de-potentiation in the STDP rule. For a given input oscillation, the balance of potentiation and de-potentiation in the STDP rule is the critical parameter that determines the phase at which an output neuron will learn to spike. These findings are robust to changes in intrinsic post-synaptic properties. Finally, we discuss implications of this mechanism for stable learning of spike-timing in the hippocampus.
PMCID: PMC3216007  PMID: 22110429
spike-timing dependent plasticity; oscillations; phase-locking; stable learning; stability of neuronal plasticity; place fields
11.  Democracy-Independence Trade-Off in Oscillating Dendrites and Its Implications for Grid Cells 
Neuron  2010;66(3):429-437.
Dendritic democracy and independence have been characterized for near-instantaneous processing of synaptic inputs. However, a wide class of neuronal computations requires input integration on long timescales. As a paradigmatic example, entorhinal grid fields have been thought to be generated by the democratic summation of independent dendritic oscillations performing direction-selective path integration. We analyzed how multiple dendritic oscillators embedded in the same neuron integrate inputs separately and determine somatic membrane voltage jointly. We found that the interaction of dendritic oscillations leads to phase locking, which sets an upper limit on the timescale for independent input integration. Factors that increase this timescale also decrease the influence that the dendritic oscillations exert on somatic voltage. In entorhinal stellate cells, interdendritic coupling dominates and causes these cells to act as single oscillators. Our results suggest a fundamental trade-off between local and global processing in dendritic trees integrating ongoing signals.
► Biophysical properties predict phase locking of oscillating dendrites ► Trade-off between dendritic independence and democracy in oscillatory regime ► Integrating inputs on long timescales favors global dendritic processing ► Intradendritic oscillations are not sufficient to maintain grid fields
PMCID: PMC3501565  PMID: 20471355
12.  The effects of cholinergic neuromodulation on neuronal phase-response curves of modeled cortical neurons 
The response of an oscillator to perturbations is described by its phase-response curve (PRC), which is related to the type of bifurcation leading from rest to tonic spiking. In a recent experimental study, we have shown that the type of PRC in cortical pyramidal neurons can be switched by cholinergic neuromodulation from type II (biphasic) to type I (monophasic). We explored how intrinsic mechanisms affected by acetylcholine influence the PRC using three different types of neuronal models: a theta neuron, single-compartment neurons and a multi-compartment neuron. In all of these models a decrease in the amount of a spike-frequency adaptation current was a necessary and sufficient condition for the shape of the PRC to change from biphasic (type II) to purely positive (type I).
PMCID: PMC2857973  PMID: 18784991
Phase response curves; Cortex; Neuromodulation; Muscarine; Acetylcholine; Pyramidal neuron; Conductance-based model; Multi-compartmental model; M-current
13.  The Role of Ongoing Dendritic Oscillations in Single-Neuron Dynamics 
PLoS Computational Biology  2009;5(9):e1000493.
The dendritic tree contributes significantly to the elementary computations a neuron performs while converting its synaptic inputs into action potential output. Traditionally, these computations have been characterized as both temporally and spatially localized. Under this localist account, neurons compute near-instantaneous mappings from their current input to their current output, brought about by somatic summation of dendritic contributions that are generated in functionally segregated compartments. However, recent evidence about the presence of oscillations in dendrites suggests a qualitatively different mode of operation: the instantaneous phase of such oscillations can depend on a long history of inputs, and under appropriate conditions, even dendritic oscillators that are remote may interact through synchronization. Here, we develop a mathematical framework to analyze the interactions of local dendritic oscillations and the way these interactions influence single cell computations. Combining weakly coupled oscillator methods with cable theoretic arguments, we derive phase-locking states for multiple oscillating dendritic compartments. We characterize how the phase-locking properties depend on key parameters of the oscillating dendrite: the electrotonic properties of the (active) dendritic segment, and the intrinsic properties of the dendritic oscillators. As a direct consequence, we show how input to the dendrites can modulate phase-locking behavior and hence global dendritic coherence. In turn, dendritic coherence is able to gate the integration and propagation of synaptic signals to the soma, ultimately leading to an effective control of somatic spike generation. Our results suggest that dendritic oscillations enable the dendritic tree to operate on more global temporal and spatial scales than previously thought; notably that local dendritic activity may be a mechanism for generating on-going whole-cell voltage oscillations.
Author Summary
A central issue in biology is how local processes yield global consequences. This is especially relevant for neurons since these spatially extended cells process local synaptic inputs to generate global action potential output. The dendritic tree of a neuron, which receives most of the inputs, expresses ion channels that can generate nonlinear dynamics. A prominent phenomenon resulting from such ion channels are voltage oscillations. The distribution of the active membrane channels throughout the cell is often highly non-uniform. This can turn the dendritic tree into a network of sparsely spaced local oscillators. Here we analyze whether local dendritic oscillators can produce cell-wide voltage oscillations. Our mathematical theory shows that indeed even when the dendritic oscillators are weakly coupled, they lock their phases and give global oscillations. We show how the biophysical properties of the dendrites determine the global locking and how it can be controlled by synaptic inputs. As a consequence of global locking, even individual synaptic inputs can affect the timing of action potentials. In fact, dendrites locking in synchrony can lead to sustained firing of the cell. We show that dendritic trees can be bistable, with dendrites locking in either synchrony or asynchrony, which may provide a novel mechanism for single cell-based memory.
PMCID: PMC2725317  PMID: 19730677
14.  Inhibition of rhythmic neural spiking by noise: the occurrence of a minimum in activity with increasing noise 
Die Naturwissenschaften  2009;96(9):1091-1097.
The effects of noise on neuronal dynamical systems are of much current interest. Here, we investigate noise-induced changes in the rhythmic firing activity of single Hodgkin–Huxley neurons. With additive input current, there is, in the absence of noise, a critical mean value µ = µc above which sustained periodic firing occurs. With initial conditions as resting values, for a range of values of the mean µ near the critical value, we have found that the firing rate is greatly reduced by noise, even of quite small amplitudes. Furthermore, the firing rate may undergo a pronounced minimum as the noise increases. This behavior has the opposite character to stochastic resonance and coherence resonance. We found that these phenomena occurred even when the initial conditions were chosen randomly or when the noise was switched on at a random time, indicating the robustness of the results. We also examined the effects of conductance-based noise on Hodgkin–Huxley neurons and obtained similar results, leading to the conclusion that the phenomena occur across a wide range of neuronal dynamical systems. Further, these phenomena will occur in diverse applications where a stable limit cycle coexists with a stable focus.
PMCID: PMC2727367  PMID: 19513592
Neuronal dynamics; Hodgkin–Huxley model; Stochastic processes; Inverse stochastic resonance
15.  Computational disease modeling – fact or fiction? 
BMC Systems Biology  2009;3:56.
Biomedical research is changing due to the rapid accumulation of experimental data at an unprecedented scale, revealing increasing degrees of complexity of biological processes. Life Sciences are facing a transition from a descriptive to a mechanistic approach that reveals principles of cells, cellular networks, organs, and their interactions across several spatial and temporal scales. There are two conceptual traditions in biological computational-modeling. The bottom-up approach emphasizes complex intracellular molecular models and is well represented within the systems biology community. On the other hand, the physics-inspired top-down modeling strategy identifies and selects features of (presumably) essential relevance to the phenomena of interest and combines available data in models of modest complexity.
The workshop, "ESF Exploratory Workshop on Computational disease Modeling", examined the challenges that computational modeling faces in contributing to the understanding and treatment of complex multi-factorial diseases. Participants at the meeting agreed on two general conclusions. First, we identified the critical importance of developing analytical tools for dealing with model and parameter uncertainty. Second, the development of predictive hierarchical models spanning several scales beyond intracellular molecular networks was identified as a major objective. This contrasts with the current focus within the systems biology community on complex molecular modeling.
During the workshop it became obvious that diverse scientific modeling cultures (from computational neuroscience, theory, data-driven machine-learning approaches, agent-based modeling, network modeling and stochastic-molecular simulations) would benefit from intense cross-talk on shared theoretical issues in order to make progress on clinically relevant problems.
PMCID: PMC2697138  PMID: 19497118
16.  Modeling nicotinic neuromodulation from global functional and network levels to nAChR based mechanisms 
Acta Pharmacologica Sinica  2009;30(6):681-693.
Neuromodulator action has received increasing attention in theoretical neuroscience. Yet models involving both neuronal populations dynamics at the circuit level and detailed receptor properties are only now being developed. Here we review recent computational approaches to neuromodulation, focusing specifically on acetylcholine (ACh) and nicotine. We discuss illustrative examples of models ranging from functional top-down to neurodynamical bottom-up. In the top-down approach, a computational theory views ACh as encoding the uncertainty expected in an environment. A different line of models accounts for neural population dynamics treating ACh as toggling neuronal networks between read-in of information and recall of memory. Building on the neurodynamics idea we discuss two models of nicotine's action with increasing degree of biological realism. Both consider explicitly receptor-level mechanisms but with different scales of detail. The first is a large-scale model of nicotine-dependent modulation of dopaminergic signaling that is capable of simulating nicotine self-administration. The second is a novel approach where circuit-level neurodynamics of the ventral tegmental area (VTA) are combined with explicit models of the dynamics of specific nicotinic ACh receptor subtypes. We show how the model is constructed based on local anatomy, electrophysiology and receptor properties and provide an illustration of its potential. In particular, we show how the model can shed light on the specific mechanisms by which nicotine controls dopaminergic neurotransmission in the VTA. This model serves us to conclude that detailed accounts for neuromodulator action at the basis of behavioral and cognitive models are crucial to understand how neuromodulators mediate their functional properties.
PMCID: PMC4002372  PMID: 19498415
mean-field model; nAChR kinetics; nicotine; computational model
17.  Cholinergic Neuromodulation Changes Phase Response Curve Shape and Type in Cortical Pyramidal Neurons 
PLoS ONE  2008;3(12):e3947.
Spike generation in cortical neurons depends on the interplay between diverse intrinsic conductances. The phase response curve (PRC) is a measure of the spike time shift caused by perturbations of the membrane potential as a function of the phase of the spike cycle of a neuron. Near the rheobase, purely positive (type I) phase-response curves are associated with an onset of repetitive firing through a saddle-node bifurcation, whereas biphasic (type II) phase-response curves point towards a transition based on a Hopf-Andronov bifurcation. In recordings from layer 2/3 pyramidal neurons in cortical slices, cholinergic action, consistent with down-regulation of slow voltage-dependent potassium currents such as the M-current, switched the PRC from type II to type I. This is the first report showing that cholinergic neuromodulation may cause a qualitative switch in the PRCs type implying a change in the fundamental dynamical mechanism of spike generation.
PMCID: PMC2596483  PMID: 19079601
18.  Cortical Pyramidal Cells as Non-Linear Oscillators: Experiment and Spike-Generation Theory 
Brain research  2007;1171:122-137.
Cortical neurons are capable of generating trains of action potentials in response to current injections. These discharges can take different forms, e.g. repetive firing that adapts during the period of current injection or bursting behaviors. We have used a combined experimental and computational approach to characterize the dynamics leading to action potential responses in single neurons. Specifically we investigated the origin of complex firing patterns in response to sinusoidal current injections. Using a reduced model, the theta neuron, alongside recordings from cortical pyramidal cells we show that both real and simulated neurons show phase locking to sine wave stimuli up to a critical frequency, above which period skipping and 1-to-x phase locking occurs. The locking behavior follows a complex “devil’s staircase” phenomena, where locked modes are interleaved with irregular firing. We further show that the critical frequency depends on the time scale of spike generation and on the level of spike frequency adaptation. These results suggest that phase locking of neuronal responses to complex input patterns can be explained by basic properties of the spike generating machinery.
PMCID: PMC2045506  PMID: 17716635
bifurcation theory; devil’s staircase; endogenous oscillators
19.  Random perturbations of spiking activity in a pair of coupled neurons 
Theory in Biosciences  2008;127(2):135-139.
We examine the effects of stochastic input currents on the firing behaviour of two coupled Type 1 or Type 2 neurons. In Hodgkin–Huxley model neurons with standard parameters, which are Type 2, in the bistable regime, synaptic transmission can initiate oscillatory joint spiking, but white noise can terminate it. In Type 1 cells (models), typified by a quadratic integrate and fire model, synaptic coupling can cause oscillatory behaviour in excitatory cells, but Gaussian white noise can again terminate it. We locally determine an approximate basin of attraction, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{A}}},$$\end{document} of the periodic orbit and explain the firing behaviour in terms of the effects of noise on the probability of escape of trajectories from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal{A}}}.$$\end{document}
PMCID: PMC2758378  PMID: 18449590

Results 1-19 (19)