Quantifying the relative contributions of divisive and subtractive feedback to rhythm generation

Biological systems are characterized by a high number of interacting components. Determining the role of each component is difficult, addressed here in the context of biological oscillations. Rhythmic behavior can result from the interplay of positive feedback that promotes bistability between high and low activity, and slow negative feedback that switches the system between the high and low activity states. Many biological oscillators include two types of negative feedback processes: divisive (decreases the gain of the positive feedback loop) and subtractive (increases the input threshold) that both contribute to slowly move the system between the high- and low-activity states. Can we determine the relative contribution of each type of negative feedback process to the rhythmic activity? Does one dominate? Do they control the active and silent phase equally? To answer these questions we use a neural network model with excitatory coupling, regulated by synaptic depression (divisive) and cellular adaptation (subtractive feedback). We first attempt to apply standard experimental methodologies: either passive observation to correlate the variations of a variable of interest to system behavior, or deletion of a component to establish whether a component is critical for the system. We find that these two strategies can lead to contradictory conclusions, and at best their interpretive power is limited. We instead develop a computational measure of the contribution of a process, by evaluating the sensitivity of the active (high activity) and silent (low activity) phase durations to the time constant of the process. The measure shows that both processes control the active phase, in proportion to their speed and relative weight. However, only the subtractive process plays a major role in setting the duration of the silent phase. This computational method can be used to analyze the role of negative feedback processes in a wide range of biological rhythms.     

Feedback-induced gain control in stochastic spiking networks

The joint influence of recurrent feedback and noise on gain control in a network of globally coupled spiking leaky integrate-and-fire neurons is studied theoretically and numerically. The context of our work is the origin of divisive versus subtractive gain control, as mixtures of these effects are seen in a variety of experimental systems. We focus on changes in the slope of the mean firing frequency-versus-input bias (f –I) curve when the gain control signal to the cells comes from the cells’ output spikes. Feedback spikes are modeled as alpha functions that produce an additive current in the current balance equation. For generality, they occur after a fixed minimum delay. We show that purely divisive gain control, i.e. changes in the slope of the f –I curve, arises naturally with this additive negative or positive feedback, due to a linearizing actions of feedback. Negative feedback alone lowers the gain, accounting in particular for gain changes in weakly electric fish upon pharmacological opening of the feedback loop as reported by Bastian (J Neurosci 6:553–562, 1986). When negative feedback is sufficiently strong it further causes oscillatory firing patterns which produce irregularities in the f –I curve. Small positive feedback alone increases the gain, but larger amounts cause abrupt jumps to higher firing frequencies. On the other hand, noise alone in open loop linearizes the f –I curve around threshold, and produces mixtures of divisive and subtractive gain control. With both noise and feedback, the combined gain control schemes produce a primarily divisive gain control shift, indicating the robustness of feedback gain control in stochastic networks. Similar results are found when the “input” parameter is the contrast of a time-varying signal rather than the bias current. Theoretical results are derived relating the slope of the f –I curve to feedback gain and noise strength. Good agreement with simulation results are found for inhibitory and excitatory feedback. Finally, divisive feedback is also found for conductance-based feedback (shunting or excitatory) with and without noise. This article is part of a special issue on Neuronal Dynamics of Sensory Coding.     

Modulation of Oscillatory Synchronization in an Interneuronal Network under the Influence of Tonic GABA-ergic Inhibition: a Model Study

The influence of a tonic GABA-ergic current on the processes of network synchronization was examined using a computer model of the neural network with shunting GABA-ergic synapses and tonic excitation that initiated spiking. The tonic inhibitory current was characterized by two parameters, the reversal potential and the conductance introduced. We found that tonic current with a reversal potential more negative than the threshold for spike generation reduces the network spiking frequency and synchronization. A monotonic decrease in the network synchronization with augmentation of the tonic current conductance was shown. We also found that a particular range of tonic current conductance leads to a bistable character of the network dynamics. Depending on the initial conditions of the network examined, spontaneous synchronous oscillations similar to epileptiform activity could appear.     

Sodium inactivation in nerve fibers

A number of models proposed to account for the sodium conductance changes are shown to fall into two classes. The Hodgkin-Huxley (HH) model falls into a class (I) in which the conductance depends on two or more independent variables controlled by independent processes. The Mullins, Hoyt, and Goldman models fall into class II in which conductance depends directly on one variable only, a variable which is controlled by two or more coupled processes. The HH and Hoyt models are used as specific examples of the two classes. It is shown that, contrary to a recently published report, the results from double experiments can be equally well accounted for by both models. It is also shown that steady-state conditioning, or “inactivation,” curves, obtained at more than one test potential, can be used to distinguish the two models. The HH equations predict that such curves should be shifted, by very small amounts, in the hyperpolarizing direction when more depolarizing test potentials are used, while the Hoyt model predicts that they should be shifted in the depolarizing direction, by quite appreciable amounts. Several pieces of published experimental information are used as tests of these predictions, and give tentative support to the class II model. Further experiments are necessary before a definite conclusion can be reached.     

Differential control of active and silent phases in relaxation models of neuronal rhythms

Rhythmic bursting activity, found in many biological systems, serves a variety of important functions. Such activity is composed of episodes, or bursts (the active phase, AP) that are separated by quiescent periods (the silent phase, SP). Here, we use mean field, firing rate models of excitatory neural network activity to study how AP and SP durations depend on two critical network parameters that control network connectivity and cellular excitability. In these models, the AP and SP correspond to the network's underlying bistability on a fast time scale due to rapid recurrent excitatory connectivity. Activity switches between the AP and SP because of two types of slow negative feedback: synaptic depression—which has a divisive effect on the network input/output function, or cellular adaptation—a subtractive effect on the input/output function. We show that if a model incorporates the divisive process (regardless of the presence of the subtractive process), then increasing cellular excitability will speed up the activity, mostly by decreasing the silent phase. Reciprocally, if the subtractive process is present, increasing the excitatory connectivity will slow down the activity, mostly by lengthening the active phase. We also show that the model incorporating both slow processes is less sensitive to parameter variations than the models with only one process. Finally, we note that these network models are formally analogous to a type of cellular pacemaker and thus similar results apply to these cellular pacemakers. Action Editor: Misha Tsodyks     

Characterization of the inward-rectifying potassium current in cat ventricular myocytes

Whole-cell membrane currents were measured in isolated cat ventricular myocytes using a suction-electrode voltage-clamp technique. An inward-rectifying current was identified that exhibited a time-dependent activation. The peak current appeared to have a linear voltage dependence at membrane potentials negative to the reversal potential. Inward current was sensitive to K channel blockers. In addition, varying the extracellular K+ concentration caused changes in the reversal potential and slope conductance expected for a K+ current. The voltage dependence of the chord conductance exhibited a sigmoidal relationship, increasing at more negative membrane potentials. Increasing the extracellular K+ concentration increased the maximal level of conductance and caused a shift in the relationship that was directly proportional to the change in reversal potential. Activation of the current followed a monoexponential time course, and the time constant of activation exhibited a monoexponential dependence on membrane potential. Increasing the extracellular K+ concentration caused a shift of this relationship that was directly proportional to the change in reversal potential. Inactivation of inward current became evident at more negative potentials, resulting in a negative slope region of the steady state current-voltage relationship between -140 and -180 mV. Steady state inactivation exhibited a sigmoidal voltage dependence, and recovery from inactivation followed a monoexponential time course. Removing extracellular Na+ caused a decrease in the slope of the steady state current-voltage relationship at potentials negative to -140 mV, as well as a decrease of the conductance of inward current. It was concluded that this current was IK1, the inward-rectifying K+ current found in multicellular cardiac preparations. The K+ and voltage sensitivity of IK1 activation resembled that found for the inward-rectifying K+ currents in frog skeletal muscle and various egg cell preparations. Inactivation of IK1 in isolated ventricular myocytes was viewed as being the result of two processes: the first involves a voltage-dependent change in conductance; the second involves depletion of K+ from extracellular spaces. The voltage-dependent component of inactivation was associated with the presence of extracellular Na+.     

Bifurcation analysis of a two-compartment hippocampal pyramidal cell model

The Pinsky-Rinzel model is a non-smooth 2-compartmental CA3 pyramidal cell model that has been used widely within the field of neuroscience. Here we propose a modified (smooth) system that captures the qualitative behaviour of the original model, while allowing the use of available, numerical continuation methods to perform full-system bifurcation and fast-slow analysis. We study the bifurcation structure of the full system as a function of the applied current and the maximal calcium conductance. We identify the bifurcations that shape the transitions between resting, bursting and spiking behaviours, and which lead to the disappearance of bursting when the calcium conductance is reduced. Insights gained from this analysis, are then used to firstly illustrate how the irregular spiking activity found between bursting and stable spiking states, can be influenced by phase differences in the calcium and dendritic voltage, which lead to corresponding changes in the calcium-sensitive potassium current. Furthermore, we use fast-slow analysis to investigate the mechanisms of bursting and show that bursting in the model is dependent on the intermediately slow variable, calcium, while the other slow variable, the activation gate of the afterhyperpolarisation current, does not contribute to setting the intraburst dynamics but participates in setting the interburst interval. Finally, we discuss how some of the described bifurcations affect spiking behaviour, during sharp-wave ripples, in a larger network of Pinsky-Rinzel cells.     

A homeostatic model of neuronal firing governed by feedback signals from the extracellular matrix

Molecules of the extracellular matrix (ECM) can modulate the efficacy of synaptic transmission and neuronal excitability. These mechanisms are crucial for the homeostatic regulation of neuronal firing over extended timescales. In this study, we introduce a simple mathematical model of neuronal spiking balanced by the influence of the ECM. We consider a neuron receiving random synaptic input in the form of Poisson spike trains and the ECM, which is modeled by a phenomenological variable involved in two feedback mechanisms. One feedback mechanism scales the values of the input synaptic conductance to compensate for changes in firing rate. The second feedback accounts for slow fluctuations of the excitation threshold and depends on the ECM concentration. We show that the ECM-mediated feedback acts as a robust mechanism to provide a homeostatic adjustment of the average firing rate. Interestingly, the activation of feedback mechanisms may lead to a bistability in which two different stable levels of average firing rates can coexist in a spiking network. We discuss the mechanisms of the bistability and how they may be related to memory function.     

The slow inward current, isi, in the rabbit sino-atrial node investigated by voltage clamp and computer simulation

The properties of the slow inward current, isi, in the sino-atrial (s.a.) node of the rabbit have been investigated using two microelectrodes to apply voltage clamp to small, spontaneously beating, preparations. Many of the experimental results can be closely simulated using the computer model of s.a. node electrical activity (Noble & Noble 1984) which has been developed from models of Purkinje fibre activity (Noble 1962; DiFrancesco & Noble 1984). Comparison of the computed reconstructions with experimental results provides a test of the validity of the modelling. Experiments using paired depolarizing clamp pulses show that inactivation of isi is calcium-entry dependent although, unlike the inactivation of Ca2+ currents in some other systems, it also shows some voltage-dependence. Re-availability (recovery from inactivation) of isi in s.a. node is much slower than inactivation at the same potential, showing that isi is not controlled by a single first order process. This very slow recovery from inactivation of isi in the s.a. node and the slow time course of its activation and inactivation at voltages near threshold (-40 to -50 mV) can be closely modelled by assuming that there are two components of 'total isi': a fast inward current, iCa,f' representing the 'gated' fraction and a second, slower, inward current component, iNaCa which, we propose, is caused by the sodium-calcium exchange that ensues when the initial Ca2+ -entry triggers the release of stored intracellular Ca2+. When repetitive trains of clamp pulses are given, a 'staircase' of isi magnitude is seen which can be increasing ('positive') or decreasing ('negative') according to the potential level and frequency of the pulse train given. When computer reconstructions of such staircases are made, it is found that the positive staircases (which, in contrast to negative staircases, imply that more complex processes than simple inactivation are present) can be closely simulated by a model which incorporates slower processes (suggested Na-Ca exchange current) in the total isi in addition to the gated current component.(ABSTRACT TRUNCATED AT 400 WORDS)     

Modelling action potentials and membrane currents of mammalian skeletal muscle fibres in coherence with potassium concentration changes in the T-tubular system

During prolonged activity the action potentials of skeletal muscle fibres change their shape. A model study was made as to whether potassium accumulation and removal in the tubular space is important with respect to those variations. Classical Hodgkin-Huxley type sodium and (potassium) delayed rectifier currents were used to determine the sarcolemmal and tubular action potentials. The resting membrane potential was described with a chloride conductance, a potassium conductance (inward rather than outward rectifier) and a sodium conductance (minor influence) in both sarcolemmal and tubular membranes. The two potassium conductances, the Na-K pump and the potassium diffusion between tubular compartments and to the external medium contributed to the settlement of the potassium concentration in the tubular space. This space was divided into 20 coupled concentric compartments. In the longitudinal direction the fibre was a cable series of 56 short segments. All the results are concerned with one of the middle segments. During action potentials, potassium accumulates in the tubular space by outward current through both the delayed and inward rectifier potassium conductances. In between the action potentials the potassium concentration decreases in all compartments owing to potassium removal processes. In the outer tubular compartment the diffusion-driven potassium export to the bathing solution is the main process. In the inner tubular compartment, potassium removal is mainly effected by re-uptake into the sarcoplasm by means of the inward rectifier and the Na-K pump. This inward transport of potassium strongly reduces the positive shift of the tubular resting membrane potential and the consequent decrease of the action potential amplitude caused by inactivation of the sodium channels. Therefore, both potassium removal processes maintain excitability of the tubular membrane in the centre of the fibre, promote excitation-contraction coupling and contribute to the prevention of fatigue. Received: 5 May 1998 / Revised version: 27 October 1998 / Accepted: 19 January 1999     

The Squid Giant Axon: Mathematical Models

The voltage clamp results of Hodgkin and Huxley have been reanalyzed in terms of alternative mathematical models. The model used for the potassium conductance changes is similar to that of the HH model except that an empirical functional relationship replaces the fourth power Law used by HH and the twenty-fifth power law used by Cole and Moore. The model used for the sodium conductance changes involves the explicit use of one variable only rather than the two variables m and h of HH. The rise and fall of the sodium conductance during a depolarizing voltage clamp is obtained by specifying that this one variable satisfies a second order differential equation which results from the coupling of two first order equations. Not only can the adjustable parameters of these models be made to give good fit to the clamp conductance data but the models can also then be used to compute action potential curves. Theoretical interpretations can also be given to these mathematical models.     

A model study of stability and oscillations in the myocardial cell membrane

As a step towards an improved understanding of cardiac arrhythmias caused by abnormal automaticity, we perform a stability analysis of a Hodgkin-Huxley model of the myocardial cell membrane (modified Beeler-Reuter, MBR). The bifurcation structure of the model is obtained as a function of three parameters: the intensity of an applied constant current; the potassium equilibrium potential representing the accumulation of K+ ions in the external medium; and the maximum conductance of the slow inward current mimicking the local application of catecholamines on the membrane. For a range of parameter values, the model exhibits either stable automaticity or bistability between two quiescent states or between a quiescent state and an oscillatory state. These transformations of the bifurcation structure are shown to depend on the interrelationship between three elements: the activation of the slow inward current, the region of high slope conductance of the time-independent potassium current functions, and the slow variables controlling the activation of the potassium current and the inactivation of the slow inward current. Reduced two- and three-dimensional models are shown to reproduce the main stability properties of the full MBR model and to facilitate the understanding of its dynamic behavior. The onset of instability and the oscillatory features of the MBR model are in good agreement with relevant experimental results, and possible sources of disagreement on certain points are discussed.     

Bradykinin inhibits potassium (M) currents in N1E-115 neuroblastoma cells. Responses resemble those in NG108-15 neuroblastoma x glioma hybrid cells

Application of bradykinin to voltage-clamped N1E-115 mouse neuroblastoma cells evoked sequential outward and inward membrane currents, accompanied by an increase and decrease of membrane conductance, respectively. Methacholine produced an inward current with a decreased conductance. The outward current response to bradykinin was imitated by intracellular inositol 1,4,5-trisphosphate (IP3). Bath application of phorbol dibutyrate induced an inward current and potentiated the response to IP3. We conclude that the response of these cells to bradykinin is identical to that of NG108-15 hybrid cells, and therefore may be attributed to the dual effects of inositol trisphosphate and diacylglycerol formed by hydrolysis of phosphatidylinositide.     

Opioids activate both an inward rectifier and a novel voltage-gated potassium conductance in the hippocampal formation

Opioid receptors were found to activate two different types of membrane potassium conductance in acutely dissociated neurons from the CA1/subiculum regions of the adult rat hippocampal formation. Opioid-responsive neurons were distinguished based on their morphology and electrophysiological responses. In one population of neurons having a multipolar, nonpyramidal cell shape, mu-selective opioid agonists increased an inward rectifying potassium current. Opioid activation of the inward rectifying conductance resulted in small outward potassium currents at resting membrane potentials and increased inward currents at hyperpolarized potentials. In a second population of nonpyramidal neurons, mu opioid agonists increased a novel voltage-gated potassium current. This current was blocked by internal CsCl2, unaffected by external BaCl2 or CdCl2, irreversibly activated by intracellular GTP-gamma-S, and inactivated by sustained depolarization. In contrast to the inward rectifying conductance, the voltage-gated conductance was not activated at resting membrane potentials or hyperpolarized potentials. The opioid-activated, voltage-gated conductance represents a new class of G protein-regulated potassium current in the brain.     

Human and Machine Learning in Non-Markovian Decision Making

Humans can learn under a wide variety of feedback conditions. Reinforcement learning (RL), where a series of rewarded decisions must be made, is a particularly important type of learning. Computational and behavioral studies of RL have focused mainly on Markovian decision processes, where the next state depends on only the current state and action. Little is known about non-Markovian decision making, where the next state depends on more than the current state and action. Learning is non-Markovian, for example, when there is no unique mapping between actions and feedback. We have produced a model based on spiking neurons that can handle these non-Markovian conditions by performing policy gradient descent [1]. Here, we examine the model’s performance and compare it with human learning and a Bayes optimal reference, which provides an upper-bound on performance. We find that in all cases, our population of spiking neurons model well-describes human performance.     

Modeling the leech heartbeat elemental oscillator I. Interactions of intrinsic and synaptic currents

We have developed a biophysical model of a pair of reciprocally inhibitory interneurons comprising an elemental heartbeat oscillator of the leech. We incorporate various intrinsic and synaptic ionic currents based on voltage-clamp data. Synaptic transmission between the interneurons consists of both a graded and a spike-mediated component. By using maximal conductances as parameters, we have constructed a canonical model whose activity appears close to the real neurons. Oscillations in the model arise from interactions between synaptic and intrinsic currents. The inhibitory synaptic currents hyperpolarize the cell, resulting in activation of a hyperpolarization-activated inward currentI _h and the removal of inactivation from regenerative inward currents. These inward currents depolarize the cell to produce spiking and inhibit the opposite cell. Spike-mediated IPSPs in the inhibited neuron cause inactivation of low-threshold Ca⁺⁺ currents that are responsible for generating the graded synaptic inhibition in the opposite cell. Thus, although the model cells can potentially generate large graded IPSPs, synaptic inhibition during canonical oscillations is dominated by the spike-mediated component.     

Current-dependent inactivation induced by sodium depletion in normal and batrachotoxin-treated frog node of Ranvier

In batrachotoxin (BTX)-treated frog node of Ranvier, in spite of a marked reduction in Na inactivation, the Na current still presents a time- and voltage-dependent inactivation that could induce a 50-60% decrease in the current. The inactivation was found to be modified by changing the amplitude of a conditioning pulse, adding tetrodotoxin in the external solution, or replacing NaCl with KCl in the external solution. Conditioning pulses were able to alter the reversal potential of the BTX-modified Na current (Vrev). Vrev was shifted toward negative values for inward conditioning currents and was shifted toward positive values for outward conditioning currents. The change in Vrev was proportional to the conditioning current amplitude. Large inward currents induced 15-25 mV shifts of Vrev. During a 10-20-ms depolarizing pulse, the inactivation and change in Vrev were proportional to the time integral of the current. For longer depolarizations, Vrev reached a steady state level proportional to the current amplitude. The conductance, as calculated from the current and the actual Vrev, showed an inactivation proportional to exp(Vrev F/RT). These observations suggest that the BTX-modified Na current induces a decrease in local Na concentrations, which results in an alteration of the driving force and the conductance. During a pulse that induced a large inward current, the Na space concentration [( Na]s) changed from 114 to 50-60 mM. In normal fibers, the reversal potential of Na current was also shifted toward negative values by a prepulse that induced a large inward current. The change in Vrev reached 5-15 mV, which corresponded to a decrease in [Na]s of 20-50 mM. This change in Vrev slightly altered the time course of Na current. On the basis of a three- compartment model (axoplasm-perinodal space-bulk solution), a Na permeability of the barrier between the space and the bulk solution (PNa,s) and a mean thickness of the space (theta) were calculated. The mean value of PNa,s was 0.0051 cm X s-1 in both normal and BTX-treated fibers, whereas the value of theta was 0.29 micron in BTX-treated fibers and 0.05 micron in normal fibers. When compared with the values calculated during K accumulation, PNa,s was 10 times smaller than PK,s and theta Na-BTX was equal to theta K.(ABSTRACT TRUNCATED AT 400 WORDS)     

Spikes annihilation in the Hodgkin-Huxley neuron

The Hodgkin–Huxley (HH) neuron is a nonlinear system with two stable states: A fixed point and a limit cycle. Both of them co-exist. The behavior of this neuron can be switched between these two equilibria, namely spiking and resting respectively, by using a perturbation method. The change from spiking to resting is named Spike Annihilation, and the transition from resting to spiking is named Spike Generation. Our intention is to determine if the HH neuron in 2D is controllable (i.e., if it can be driven from a quiescent state to a spiking state and vice versa). It turns out that the general system is unsolvable.¹ In this paper, first of all,² we analytically prove the existence of a brief current pulse, which, when delivered to the HH neuron during its repetitively firing state, annihilates its spikes. We also formally derive the characteristics of this brief current pulse. We then proceed to explore experimentally, by using numerical simulations, the properties of this pulse, namely the range of time when it can be inserted (the minimum phase and the maximum phase), its magnitude, and its duration. In addition, we study the solution of annihilating the spikes by using two successive stimuli, when the first is, of its own, unable to annihilate the neuron. Finally, we investigate the inverse problem of annihilation, namely the spike generation problem, when the neuron switches from resting to firing. ¹ This conclusion is a consequence of three well-known fundamental results, namely Hilbert 16th Problem, the Poincare–Bendixon Theorem and the Hopf Bifurcation Theorem. ² We are extremely grateful to the feedback we received from the anonymous Referees to the initial version of the paper. Their comments significantly improved the quality of the current version. Thanks a lot!     

Behavior of delayed current under voltage clamp in the supramedullary neurons of puffer

Depolarizations applied to voltage-clamped cells bathed in the normal solution disclose an initial inward current followed by a delayed outward current. The maximum slope conductance for the peak initial current is about 30 times the leak conductance, but the maximum slope conductance for the delayed current is only about 10 times the leak conductance. During depolarizations for as long as 30 sec, the outward current does not maintain a steady level, but declines first exponentially with a time constant of about 6 msec; it then tends to increase for the next few seconds; finally, it declines slowly with a half-time of about 5 sec. Concomitant with the changes of the outward current, the membrane conductance changes, although virtually no change in electromotive force occurs. Thus, the changes in the membrane conductance represent two phases of K inactivation, one rapidly developing, the other slowly occurring, and a phase of K reactivation, which is interposed between the two inactivations. In isosmotic KCl solution after a conditioning hyperpolarization there occurs an increase in K permeability upon depolarization. When the depolarizations are maintained, the increase of K permeability undergoes changes similar to those observed in the normal medium. The significance of the K inactivation is discussed in relation to the after-potential of the nerve cells.