Frequency tuning of input-output relation in a rat cortical neuron in-vitro

Dynamic clamp constructed phase diagram for the Hodgkin and Huxley model of excitability

Excitability-a threshold-governed transient in transmembrane voltage-is a fundamental physiological process that controls the function of the heart, endocrine, muscles, and neuronal tissues. The 1950s Hodgkin and Huxley explicit formulation provides a mathematical framework for understanding excitability, as the consequence of the properties of voltage-gated sodium and potassium channels. The Hodgkin-Huxley model is more sensitive to parametric variations of protein densities and kinetics than biological systems whose excitability is apparently more robust. It is generally assumed that the model's sensitivity reflects missing functional relations between its parameters or other components present in biological systems. Here we experimentally assembled excitable membranes using the dynamic clamp and voltage-gated potassium ionic channels (Kv1.3) expressed in Xenopus oocytes. We take advantage of a theoretically derived phase diagram, where the phenomenon of excitability is reduced to two dimensions defined as combinations of the Hodgkin-Huxley model parameters, to examine functional relations in the parameter space. Moreover, we demonstrate activity dependence and hysteretic dynamics over the phase diagram due to the impacts of complex slow inactivation kinetics. The results suggest that maintenance of excitability amid parametric variation is a low-dimensional, physiologically tenable control process. In the context of model construction, the results point to a potentially significant gap between high-dimensional models that capture the full measure of complexity displayed by ion channel function and the lower dimensionality that captures physiological function.

Cellular function given parametric variation in the Hodgkin and Huxley model of excitability

How is reliable physiological function maintained in cells despite considerable variability in the values of key parameters of multiple interacting processes that govern that function? Here, we use the classic Hodgkin-Huxley formulation of the squid giant axon action potential to propose a possible approach to this problem. Although the full Hodgkin-Huxley model is very sensitive to fluctuations that independently occur in its many parameters, the outcome is in fact determined by simple combinations of these parameters along two physiological dimensions: structural and kinetic (denoted S and K, respectively). Structural parameters describe the properties of the cell, including its capacitance and the densities of its ion channels. Kinetic parameters are those that describe the opening and closing of the voltage-dependent conductances. The impacts of parametric fluctuations on the dynamics of the system-seemingly complex in the high-dimensional representation of the Hodgkin-Huxley model-are tractable when examined within the S-K plane. We demonstrate that slow inactivation, a ubiquitous activity-dependent feature of ionic channels, is a powerful local homeostatic control mechanism that stabilizes excitability amid changes in structural and kinetic parameters.

Universality, complexity and the praxis of biology: Two case studies

The phenomenon of biology provides a prime example for a naturally occurring complex system. The approach to this complexity reflects the tension between a reductionist, reverse-engineering stance, and more abstract, systemic ones. Both of us are reductionists, but our observations challenge reductionism, at least the naive version of it. Here we describe the challenge, focusing on two universal characteristics of biological complexity: two-way microscopic-macroscopic degeneracy, and lack of time scale separation within and between levels of organization. These two features and their consequences for the praxis of experimental biology, reflect inherent difficulties in separating the dynamics of any given level of organization from the coupled dynamics of all other levels, including the environment within which the system is embedded. Where these difficulties are not deeply acknowledged, the impacts of fallacies that are inherent to naive reductionism are significant. In an era where technology enables experimental high-resolution access to numerous observables, the challenge faced by the mature reductionist-identification of relevant microscopic variables-becomes more demanding than ever. The demonstrations provided here are taken from two very different biological realizations: populations of microorganisms and populations of neurons, thus making the lesson potentially general.

A computational paradigm for dynamic logic-gates in neuronal activity

In 1943 McCulloch and Pitts suggested that the brain is composed of reliable logic-gates similar to the logic at the core of today's computers. This framework had a limited impact on neuroscience, since neurons exhibit far richer dynamics. Here we propose a new experimentally corroborated paradigm in which the truth tables of the brain's logic-gates are time dependent, i.e., dynamic logic-gates (DLGs). The truth tables of the DLGs depend on the history of their activity and the stimulation frequencies of their input neurons. Our experimental results are based on a procedure where conditioned stimulations were enforced on circuits of neurons embedded within a large-scale network of cortical cells in-vitro. We demonstrate that the underlying biological mechanism is the unavoidable increase of neuronal response latencies to ongoing stimulations, which imposes a non-uniform gradual stretching of network delays. The limited experimental results are confirmed and extended by simulations and theoretical arguments based on identical neurons with a fixed increase of the neuronal response latency per evoked spike. We anticipate our results to lead to better understanding of the suitability of this computational paradigm to account for the brain's functionalities and will require the development of new systematic mathematical methods beyond the methods developed for traditional Boolean algebra.

Is the brain's inertia for motor movements different for acceleration and deceleration?

The brain's ability to synchronize movements with external cues is used daily, yet neuroscience is far from a full understanding of the brain mechanisms that facilitate and set behavioral limits on these sequential performances. This functional magnetic resonance imaging (fMRI) study was designed to help understand the neural basis of behavioral performance differences on a synchronizing movement task during increasing (acceleration) and decreasing (deceleration) metronome rates. In the MRI scanner, subjects were instructed to tap their right index finger on a response box in synchrony to visual cues presented on a display screen. The tapping rate varied either continuously or in discrete steps ranging from 0.5 Hz to 3 Hz. Subjects were able to synchronize better during continuously accelerating rhythms than in continuously or discretely decelerating rhythms. The fMRI data revealed that the precuneus was activated more during continuous deceleration than during acceleration with the hysteresis effect significant at rhythm rates above 1 Hz. From the behavioral data, two performance measures, tapping rate and synchrony index, were derived to further analyze the relative brain activity during acceleration and deceleration of rhythms. Tapping rate was associated with a greater brain activity during deceleration in the cerebellum, superior temporal gyrus and parahippocampal gyrus. Synchrony index was associated with a greater activity during the continuous acceleration phase than during the continuous deceleration or discrete acceleration phases in a distributed network of regions including the prefrontal cortex and precuneus. These results indicate that the brain's inertia for movement is different for acceleration and deceleration, which may have implications in understanding the origin of our perceptual and behavioral limits.

Dopamine modulation of Ih improves temporal fidelity of spike propagation in an unmyelinated axon

We studied how conduction delays of action potentials in an unmyelinated axon depended on the history of activity and how this dependence was changed by the neuromodulator dopamine (DA). The pyloric dilator axons of the stomatogastric nervous system in the lobster, Homarus americanus, exhibited substantial activity-dependent hyperpolarization and changes in spike shape during repetitive activation. The conduction delays varied by several milliseconds per centimeter, and, during activation with realistic burst patterns or Poisson-like patterns, changes in delay occurred over multiple timescales. The mean delay increased, whereas the resting membrane potential hyperpolarized with a time constant of several minutes. Concomitantly with the mean delay, the variability of delay also increased. The variability of delay was not a linear or monotonic function of instantaneous spike frequency or spike shape parameters, and the relationship between these parameters changed with the increase in mean delay. Hyperpolarization was counteracted by a hyperpolarization-activated inward current (I(h)), and the magnitude of I(h) critically determined the temporal fidelity of spike propagation. Pharmacological block of I(h) increased the change in delay and the variability of delay, and increasing I(h) by application of DA diminished both. Consequently, the temporal fidelity of pattern propagation was substantially improved in DA. Standard measurements of changes in excitability or delay with paired stimuli or tonic stimulation failed to capture the dynamics of spike conduction. These results indicate that spike conduction can be extremely sensitive to the history of axonal activity and to the presence of neuromodulators, with potentially important consequences for temporal coding.

Beyond faithful conduction: short-term dynamics, neuromodulation, and long-term regulation of spike propagation in the axon

Most spiking neurons are divided into functional compartments: a dendritic input region, a soma, a site of action potential initiation, an axon trunk and its collaterals for propagation of action potentials, and distal arborizations and terminals carrying the output synapses. The axon trunk and lower order branches are probably the most neglected and are often assumed to do nothing more than faithfully conducting action potentials. Nevertheless, there are numerous reports of complex membrane properties in non-synaptic axonal regions, owing to the presence of a multitude of different ion channels. Many different types of sodium and potassium channels have been described in axons, as well as calcium transients and hyperpolarization-activated inward currents. The complex time- and voltage-dependence resulting from the properties of ion channels can lead to activity-dependent changes in spike shape and resting potential, affecting the temporal fidelity of spike conduction. Neural coding can be altered by activity-dependent changes in conduction velocity, spike failures, and ectopic spike initiation. This is true under normal physiological conditions, and relevant for a number of neuropathies that lead to abnormal excitability. In addition, a growing number of studies show that the axon trunk can express receptors to glutamate, GABA, acetylcholine or biogenic amines, changing the relative contribution of some channels to axonal excitability and therefore rendering the contribution of this compartment to neural coding conditional on the presence of neuromodulators. Long-term regulatory processes, both during development and in the context of activity-dependent plasticity may also affect axonal properties to an underappreciated extent.

Dynamics of excitability over extended timescales in cultured cortical neurons

Although neuronal excitability is well understood and accurately modeled over timescales of up to hundreds of milliseconds, it is currently unclear whether extrapolating from this limited duration to longer behaviorally relevant timescales is appropriate. Here we used an extracellular recording and stimulation paradigm that extends the duration of single-neuron electrophysiological experiments, exposing the dynamics of excitability in individual cultured cortical neurons over timescales hitherto inaccessible. We show that the long-term neuronal excitability dynamics is unstable and dominated by critical fluctuations, intermittency, scale-invariant rate statistics, and long memory. These intrinsic dynamics bound the firing rate over extended timescales, contrasting observed short-term neuronal response to stimulation onset. Furthermore, the activity of a neuron over extended timescales shows transitions between quasi-stable modes, each characterized by a typical response pattern. Like in the case of rate statistics, the short-term onset response pattern that often serves to functionally define a given neuron is not indicative of its long-term ongoing response. These observations question the validity of describing neuronal excitability based on temporally restricted electrophysiological data, calling for in-depth exploration of activity over wider temporal scales. Such extended experiments will probably entail a different kind of neuronal models, accounting for the unbounded range, from milliseconds up.

Adaptive transition rates in excitable membranes

Adaptation of activity in excitable membranes occurs over a wide range of timescales. Standard computational approaches handle this wide temporal range in terms of multiple states and related reaction rates emanating from the complexity of ionic channels. The study described here takes a different (perhaps complementary) approach, by interpreting ion channel kinetics in terms of population dynamics. I show that adaptation in excitable membranes is reducible to a simple Logistic-like equation in which the essential non-linearity is replaced by a feedback loop between the history of activation and an adaptive transition rate that is sensitive to a single dimension of the space of inactive states. This physiologically measurable dimension contributes to the stability of the system and serves as a powerful modulator of input–output relations that depends on the patterns of prior activity; an intrinsic scale free mechanism for cellular adaptation that emerges from the microscopic biophysical properties of ion channels of excitable membranes.

Order-based representation in random networks of cortical neurons

The wide range of time scales involved in neural excitability and synaptic transmission might lead to ongoing change in the temporal structure of responses to recurring stimulus presentations on a trial-to-trial basis. This is probably the most severe biophysical constraint on putative time-based primitives of stimulus representation in neuronal networks. Here we show that in spontaneously developing large-scale random networks of cortical neurons in vitro the order in which neurons are recruited following each stimulus is a naturally emerging representation primitive that is invariant to significant temporal changes in spike times. With a relatively small number of randomly sampled neurons, the information about stimulus position is fully retrievable from the recruitment order. The effective connectivity that makes order-based representation invariant to time warping is characterized by the existence of stations through which activity is required to pass in order to propagate further into the network. This study uncovers a simple invariant in a noisy biological network in vitro; its applicability under in vivo constraints remains to be seen.

Low-frequency stimulation induces stable transitions in stereotypical activity in cortical networks

Reverberating spontaneous synchronized brain activity is believed to play an important role in neural information processing. Whether and how external stimuli can influence this spontaneous activity is poorly understood. Because periodic synchronized network activity is also prominent in in vitro neuronal cultures, we used cortical cultures grown on multielectrode arrays to examine how spontaneous activity is affected by external stimuli. Spontaneous network activity before and after low-frequency electrical stimulation was quantified in several ways. Our results show that the initially stable pattern of stereotypical spontaneous activity was transformed into another activity pattern that remained stable for at least 1 h. The transformations consisted of changes in single site and culture-wide network activity as well as in the spatiotemporal dynamics of network bursting. We show for the first time that low-frequency electrical stimulation can induce long-lasting alterations in spontaneous activity of cortical neuronal networks. We discuss whether the observed transformations in network activity could represent a switch in attractor state.

Conditional firing probabilities in cultured neuronal networks: a stable underlying structure in widely varying spontaneous activity patterns

To properly observe induced connectivity changes after training sessions, one needs a network model that describes individual relationships in sufficient detail to enable observation of induced changes and yet reveals some kind of stability in these relationships. We analyzed spontaneous firing activity in dissociated rat cortical networks cultured on multi-electrode arrays by means of the conditional firing probability. For all pairs (i, j) of the 60 electrodes, we calculated conditional firing probability (CFP(i,j)[tau]) as the probability of an action potential at electrode j at t = tau, given that one was detected at electrode i at t = 0. If a CFP(i,j)[tau] distribution clearly deviated from a flat one, electrodes i and j were considered to be related. For all related electrode pairs, a function was fitted to the CFP-curve to obtain parameters for 'strength' and 'delay' (i.e. maximum and latency of the maximum of the curve) of each relationship. In young cultures the set of identified relationships changed rather quickly. At 16 days in vitro (DIV) 50% of the set changed within 2 days. Beyond 25 DIV this set stabilized: during a week more than 50% of the set remained intact. Most individual relationships developed rather gradually. Moreover, beyond 25 DIV relational strength appeared quite stable, with coefficients of variation (100 x SD/mean) around 25% in periods of approximately 10 h. CFP analysis provides a robust method to describe the underlying probabilistic structure of highly varying spontaneous activity in cultured cortical networks. It may offer a suitable basis for plasticity studies, in the case of changes in the probabilistic structure. CFP analysis monitors all pairs of electrodes instead of just a selected one. Still, it is likely to describe the network in sufficient detail to detect subtle changes in individual relationships.

History-dependent multiple-time-scale dynamics in a single-neuron model

History-dependent characteristic time scales in dynamics have been observed at several levels of organization in neural systems. Such dynamics can provide powerful means for computation and memory. At the level of the single neuron, several microscopic mechanisms, including ion channel kinetics, can support multiple-time-scale dynamics. How the temporally complex channel kinetics gives rise to dynamical properties of the neuron is not well understood. Here, we construct a model that captures some features of the connection between these two levels of organization. The model neuron exhibits history-dependent multiple-time-scale dynamics in several effects: first, after stimulation, the recovery time scale is related to the stimulation duration by a power-law scaling; second, temporal patterns of neural activity in response to ongoing stimulation are modulated over time; finally, the characteristic time scale for adaptation after a step change in stimulus depends on the duration of the preceding stimulus. All these effects have been observed experimentally and are not explained by current single-neuron models. The model neuron here presented is composed of an ensemble of ion channels that can wander in a large pool of degenerate inactive states and thus exhibits multiple-time-scale dynamics at the molecular level. Channel inactivation rate depends on recent neural activity, which in turn depends through modulations of the neural response function on the fraction of active channels. This construction produces a model that robustly exhibits nonexponential history-dependent dynamics, in qualitative agreement with experimental results.

Learning in ex-vivo developing networks of cortical neurons

This contribution describes the use of multi-site interaction with large cortical networks in the study of learning. The general physiological properties of the network are described, and the concept of learning is mapped to the experimental network preparation. Learning is then analyzed in terms of exploration (defined as changes in the configuration of associations within the biological network) and recognition (the stabilization of "worthy" associations).

Single-neuron discharge properties and network activity in dissociated cultures of neocortex

Cultures of neurons from rat neocortex exhibit spontaneous, temporally patterned, network activity. Such a distributed activity in vitro constitutes a possible framework for combining theoretical and experimental approaches, linking the single-neuron discharge properties to network phenomena. In this work, we addressed the issue of closing the loop, from the identification of the single-cell discharge properties to the prediction of collective network phenomena. Thus, we compared these predictions with the spontaneously emerging network activity in vitro, detected by substrate arrays of microelectrodes. Therefore, we characterized the single-cell discharge properties to Gauss-distributed noisy currents, under pharmacological blockade of the synaptic transmission. Such stochastic currents emulate a realistic input from the network. The mean (m) and variance (s(2)) of the injected current were varied independently, reminiscent of the extended mean-field description of a variety of possible presynaptic network organizations and mean activity levels, and the neuronal response was evaluated in terms of the steady-state mean firing rate (f). Experimental current-to-spike-rate responses f(m, s(2)) were similar to those of neurons in brain slices, and could be quantitatively described by leaky integrate-and-fire (IF) point neurons. The identified model parameters were then used in numerical simulations of a network of IF neurons. Such a network reproduced a collective activity, matching the spontaneous irregular population bursting, observed in cultured networks. We finally interpret such a collective activity and its link with model details by the mean-field theory. We conclude that the IF model is an adequate minimal description of synaptic integration and neuronal excitability, when collective network activities are considered in vitro.

Selective adaptation in networks of cortical neurons

A key property of neural systems is their ability to adapt selectively to stimuli with different features. Using multisite electrical recordings from networks of cortical neurons developing ex vivo, we show that neurons adapt selectively to different stimuli invading the network. We focus on selective adaptation to frequent and rare stimuli; networks were stimulated at two sites with two different stimulus frequencies. When both stimuli were presented within the same period, neurons in the network attenuated their responsiveness to the more frequent input, whereas their responsiveness to the rarely delivered stimuli showed a marked average increase. The amplification of the response to rare stimuli required the presence of the other, more frequent stimulation source. By contrast, the decreased response to the frequent stimuli occurred regardless of the presence of the rare stimuli. Analysis of the response of single units suggests that both of these effects are caused by changes in synaptic transmission. By using synaptic blockers, we find that the increased responsiveness to the rarely stimulated site depends specifically on fast GABAergic transmission. Thus, excitatory synaptic depression, the inhibitory sub-network, and their balance play an active role in generating selective gain control. The observation that selective adaptation arises naturally in a network of cortical neurons developing ex vivo indicates that this is an inherent feature of spontaneously organizing cortical networks.

Modeling the process of rate selection in neuronal activity

We present the elements of a mathematical computational model that reflects the experimental finding that the time-scale of a neuron is not fixed; but rather varies with the history of its stimulus. Unlike most physiological models, there are no pre-determined rates associated with transitions between states of the system nor are there pre-determined constants associated with adaptation rates; instead, the model is a kind of "modulating automata" where the rates emerge from the history of the system itself. We focus in this paper on the temporal dynamics of a neuron and show how a simple internal structure will give rise to complex temporal behavior. The internal structure modeled here is an abstraction of a reasonably well-understood physiological structure. We also suggest that this behavior can be used to transform a "rate" code into a "temporal one".