This book explains the relationship of electrophysiology, nonlinear dynamics, and the computational properties of neurons, with each concept presented in terms of both neuroscience and mathematics and illustrated using geometrical intuition In order to model neuronal behavior or to interpret the results of modeling studies, neuroscientists must call upon methods of nonlinear dynamics This book offers an introduction to nonlinear dynamical systems theory for researchers and graduate students in neuroscience It also provides an overview of neuroscience for mathematicians who want to learn the basic facts of electrophysiology "Dynamical Systems in Neuroscience" presents a systematic study of the relationship of electrophysiology, nonlinear dynamics, and computational properties of neurons It emphasizes that information processing in the brain depends not only on the electrophysiological properties of neurons but also on their dynamical properties The book introduces dynamical systems starting with one- and two-dimensional Hodgkin-Huxley-type models and continuing to a description of bursting systems Each chapter proceeds from the simple to the complex, and provides sample problems at the end The book explains all necessary mathematical concepts using geometrical intuition; it includes many figures and few equations, making it especially suitable for non-mathematicians Each concept is presented in terms of both neuroscience and mathematics, providing a link between the two disciplines Nonlinear dynamical systems theory is at the core of computational neuroscience research, but it is not a standard part of the graduate neuroscience curriculum - or taught by math or physics department in a way that is suitable for students of biology This book offers neuroscience students and researchers a comprehensive account of concepts and methods increasingly used in computational neuroscience

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Dynamical Systems in Neuroscience

Anatomic and physiologic data are used to analyze the energy expenditure on different components of excitatory signaling in the grey matter of rodent brain. Action potentials and postsynaptic effects of glutamate are predicted to consume much of the energy (47% and 34%, respectively), with the resting potential consuming a smaller amount (13%), and glutamate recycling using only 3%. Energy usage depends strongly on action potential rate--an increase in activity of 1 action potential/cortical neuron/s will raise oxygen consumption by 145 mL/100 g grey matter/h. The energy expended on signaling is a large fraction of the total energy used by the brain; this favors the use of energy efficient neural codes and wiring patterns. Our estimates of energy usage predict the use of distributed codes, with <or=15% of neurons simultaneously active, to reduce energy consumption and allow greater computing power from a fixed number of neurons. Functional magnetic resonance imaging signals are likely to be dominated by changes in energy usage associated with synaptic currents and action potential propagation.

https://journals.sagepub.com/doi/pdf/10.1097/00004647-200110000-00001

An Energy Budget for Signaling in the Grey Matter of the Brain

Synchronous rhythms represent a core mechanism for sculpting temporal coordination of neural activity in the brain-wide network. This review focuses on oscillations in the cerebral cortex that occur during cognition, in alert behaving conditions. Over the last two decades, experimental and modeling work has made great strides in elucidating the detailed cellular and circuit basis of these rhythms, particularly gamma and theta rhythms. The underlying physiological mechanisms are diverse (ranging from resonance and pacemaker properties of single cells to multiple scenarios for population synchronization and wave propagation), but also exhibit unifying principles. A major conceptual advance was the realization that synaptic inhibition plays a fundamental role in rhythmogenesis, either in an interneuronal network or in a reciprocal excitatory-inhibitory loop. Computational functions of synchronous oscillations in cognition are still a matter of debate among systems neuroscientists, in part because the notion of regular oscillation seems to contradict the common observation that spiking discharges of individual neurons in the cortex are highly stochastic and far from being clocklike. However, recent findings have led to a framework that goes beyond the conventional theory of coupled oscillators and reconciles the apparent dichotomy between irregular single neuron activity and field potential oscillations. From this perspective, a plethora of studies will be reviewed on the involvement of long-distance neuronal coherence in cognitive functions such as multisensory integration, working memory, and selective attention. Finally, implications of abnormal neural synchronization are discussed as they relate to mental disorders like schizophrenia and autism.

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Neurophysiological and Computational Principles of Cortical Rhythms in Cognition

Bifurcation mechanisms involved in the generation of action potentials (spikes) by neurons are reviewed here. We show how the type of bifurcation determines the neuro-computational properties of the cells. For example, when the rest state is near a saddle-node bifurcation, the cell can fire all-or-none spikes with an arbitrary low frequency, it has a well-defined threshold manifold, and it acts as an integrator; i.e. the higher the frequency of incoming pulses, the sooner it fires. In contrast, when the rest state is near an Andronov–Hopf bifurcation, the cell fires in a certain frequency range, its spikes are not all-or-none, it does not have a well-defined threshold manifold, it can fire in response to an inhibitory pulse, and it acts as a resonator; i.e. it responds preferentially to a certain (resonant) frequency of the input. Increasing the input frequency may actually delay or terminate its firing. We also describe the phenomenon of neural bursting, and we use geometric bifurcation theory to extend the existing classification of bursters, including many new types. We discuss how the type of burster defines its neuro-computational properties, and we show that different bursters can interact, synchronize and process information differently.

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Neural excitability, spiking and bursting

The action potential of the squid giant axon is formed by just two voltage- dependent conductances in the cell membrane, yet mammalian central neurons typically express more than a dozen different types of voltage-dependent ion channels. This rich repertoire of channels allows neurons to encode information by generating action potentials with a wide range of shapes, frequencies and patterns. Recent work offers an increasingly detailed understanding of how the expression of particular channel types underlies the remarkably diverse firing behaviour of various types of neurons.

The action potential in mammalian central neurons

1. Using whole cell recording techniques, we studied subthreshold and suprathreshold voltage responses to oscillatory current inputs in neurons from the sensorimotor cortex of juvenile rats. 2. Based on firing patterns, neurons were classified as regular spiking (RS), intrinsic bursting (IB), and fast spiking (FS). The subthreshold voltage-current relationships of RS and IB neurons were rectifying whereas FS neurons were almost ohmic near rest. 3. Frequency response curves (FRCs) for neurons were determined by analyzing the frequency content of inputs and outputs. The FRCs of most neurons were voltage dependent at frequencies below, but not above, 20 Hz. Approximately 60% of RS and IB neurons had a membrane resonance at their resting potential. Resonant frequencies were between 0.7 and 2.5 Hz (24-26 degrees C) near -70 mV and usually increased with hyperpolarization and decreased with depolarization. The remaining RS and IB neurons and all FS neurons were nonresonant. 4. Resonant neurons near rest had a selective coupling between oscillatory inputs and firing. These neurons selectively fired action potentials when the frequency of the swept-sine-wave (ZAP) current input was near the resonant frequency. However, when these neurons were depolarized to -60 mV, spike firing was associated with many input frequencies rather than selectively near the resonant frequency. 5. We examined three subthreshold currents that could cause low-frequency resonance: IH, a slow, hyperpolarization-activated cation current that was blocked by external Cs+ but not Ba2+; IIR, an instantaneously activating, inwardly rectifying K+ current that was blocked by both Cs+ and Ba2+; and INaP, an quickly activating, inwardly rectifying persistent Na+ current that was blocked by tetrodotoxin (TTX). Voltage-clamp experiments defined the relative steady state activation ranges of these currents. IIR (activates below -80 mV) and INaP (activates above -65 mV) are unlikely to interact with each other because their activation ranges never overlap. However, both currents may interact with IH, which activated variably at potentials between -50 and -90 mV in different neurons. 6. We found that IH produces subthreshold response. Consistent with this, subthreshold resonance was blocked by external Cs+ but not Ba2+ or TTX. Application of Ba2+ enlarged FRCs and resonance at potentials below -80 mV, indicating that IK,ir normally attenuates resonance. Application of TTX greatly diminished resonance at potentials more depolarized than -65 mV, indicating that INaP normally amplifies resonance at these potentials. 7. The ZAP current input may be viewed as a model of oscillatory currents that arise in neocortical neurons during synchronized activity in the brain. We propose that the frequency selectivity endowed on neurons by IH may contribute to their participation in synchronized firing. The voltage dependence of the frequency-selective coupling between oscillatory inputs and spikes may indicate a novel mechanism for controlling the extent of low-frequency synchronized activity in the neocortex.

Subthreshold membrane resonance in neocortical neurons

1 We obtained whole cell data from sensorimotor cortical neurons of in vitro slices (juvenile rats) and observed a low-frequency resonance (1-2 Hz) in their voltage responses We constructed models of subthreshold membrane currents to determine whether a hyperpolarization-activated cation current (IH) is sufficient to account for this resonance 2 Parameter values for a basic IH (BH) model were estimated from voltage-clamp experiments at room temperature The BH model formed a component of a reduced membrane (RM) model On numerical integration, the RM model exhibited voltage sags and rebounds to injected current pulses; maximal voltage responses to injected oscillatory currents occurred near 2 Hz 3 We compared the experimentally measured frequency-response curves (FRCs) at room temperature with the theoretical FRCs derived from the RM model The theoretical FRCs exhibited resonant humps with peaks between 1 and 2 Hz At 36 degrees C, the theoretical FRCs peaked near 10 Hz The characteristics of theoretical and observed FRCs were in close agreement, demonstrating that IH is sufficient to cause resonance Thus we classified IH as a resonator current 4 We developed a technique, the reactive current clamp (RCC), to inject a computer-calculated current corresponding to a membrane ionic current in response to the membrane potential of the neuron This enabled us to study the interaction of an artificial ionic current with living neurons (electronic pharmacology or EP-method) Using the RCC, a simplified version of the BH model was used to cancel an endogenous IH (electronic antagonism) and to introduce an artificial IH (electronic expression) when an endogenous IH was absent Antagonism of IH eliminated sags and rebounds, whereas expression of IH endowed neurons with resonance and the frequency-selective firing that accompanies resonance in neurons with an endogenous IH Previous investigations have relied on the specificity of pharmacological agents to block ionic channels, eg, Cs+ to block IH However, Cs+ additionally affects other currents This represents the first time an in vitro modeling technique (RCC) has been used to antagonize a specific endogenous current, IH 5 A simplified RM model yielded values of the resonant frequency and Q (an index of the sharpness of resonance), which rose almost linearly between -55 and -80 mV Resonant frequencies could be much higher than fH = (2 pi tau H) - 1 where tau H is the activation time constant for IH 6 In the FRCs, resonance appeared as a hump at intermediate frequencies because of low- and high-frequency attenuations due to IH and membrane capacitance, respectively Changing the parameters of IH altered the low-frequency attenuation and, hence, the resonance Changes in the leak conductance affected both the low- and high-frequency attenuations 7 We modeled an inwardly rectifying K+ current (IIR) and a persistent Na+ current (INaP) to study their effects on resonance Neither current produced resonance in the absence of IH We found that IIR attenuated, whereas INaP amplified resonance Thus IIR and INaP are classified as attenuator and amplifier currents, respectively 8 Resonators and attenuators differ in that they have longer and shorter time constants, respectively, compared with the membrane time constant Therefore, an increase in the leak conductance decreases the membrane time constant, which can transform an attenuator into a resonator, altering the frequency response This suggests a novel mechanism for modulating the frequency responses of neurons to inputs 9 These investigations have provided a theoretical framework for detailed understanding of mechanisms that produce resonance in cortical neurons Resonance is one aspect of the intrinsic rhythmicity of neurons The rhythmicity due to IH resonance is latent until it is revealed by oscillatory inputs (ABSTRACT TRUNCATED)

Models of subthreshold membrane resonance in neocortical neurons

1. The whole-cell patch-clamp technique was used to study the membrane properties of identified cholinergic and noncholinergic laterodorsal tegmental neurons in slices of rat brain maintained in vitro. 2. On the basis of their expression of the transient outward potassium current IA and the transient inward calcium current IT, three classes of neurons were observed: type I neurons exhibited a large IT; type II neurons exhibited a prominent IA; and type III neurons exhibited both IA and IT. 3. Combining intracellular deposition of biocytin with NADPH diaphorase histochemistry revealed that the vast majority of type III neurons were cholinergic, whereas only a minority of type I and type II neurons were cholinergic. Thus mesopontine cholinergic neurons possess intrinsic ionic currents capable of inducing burst firing. 4. Delineation of the intrinsic membrane properties of identified mesopontine cholinergic neurons, in concert with recent results regarding the responses of these neurons to neurotransmitter agents, has led us to present a unifying and mechanistic hypothesis of brain stem cholinergic function in the control of behavioral states.

Membrane properties of mesopontine cholinergic neurons studied with the whole-cell patch-clamp technique: implications for behavioral state control

1. We constructed a mathematical model of the subthreshold electrical behavior of neurons in the nucleus mediodorsalis thalami (MDT) to elucidate the basis of a Ni(2+)-sensitive low-frequency (2-4 Hz) resonance found previously in these neurons. 2. A model that included the low- and high-threshold Ca2+ currents (IT and IL), a Ca(2+)-activated K+ current (IC), a rapidly inactivating K+ current (IA), a voltage-dependent K+ current which we call IKx, and a voltage-independent leak current (Il), successfully simulated the low-threshold spike observed in MDT neurons. This model (the MDT model) and a minimal version of the model containing only IT and I1 (the minimal MDT model) were used in the analysis. 3. An impedance function was derived for a linearized version of the MDT model. This showed that the model predicts a low-frequency (2-4 Hz) resonance in the voltage response to "small" oscillatory current inputs (producing voltage changes of 30 mV. 6. Large oscillatory current inputs evoked significantly nonlinear voltage responses in the minimal MDT model, but the 2- to 4-Hz frequency selectivity (predicted from the linearized impedance) remained. 7. We conclude that the properties of the low-threshold Ca2+ current, IT, are sufficient to explain the Ni(2+)-sensitive 2- to 4-Hz resonance seen in MDT neurons.(ABSTRACT TRUNCATED AT 400 WORDS)

B. Hutcheon

Papers

Subthreshold membrane resonance in neocortical neurons

Models of subthreshold membrane resonance in neocortical neurons

Membrane properties of mesopontine cholinergic neurons studied with the whole-cell patch-clamp technique: implications for behavioral state control

Low-threshold calcium current and resonance in thalamic neurons: a model of frequency preference

Isoflurane inhibits calcium currents in neocortical neurons