A theoretical study of neural adaptation and transient responses due to inhibitory feedback

Linear versus nonlinear signal transmission in neuron models with adaptation currents or dynamic thresholds

Spike-frequency adaptation is a prominent aspect of neuronal dynamics that shapes a neuron's signal processing properties on timescales ranging from about 10 ms to >1 s. For integrate-and-fire model neurons spike-frequency adaptation is incorporated either as an adaptation current or as a dynamic firing threshold. Whether a physiologically observed adaptation mechanism should be modeled as an adaptation current or a dynamic threshold, however, is not known. Here we show that a dynamic threshold has a divisive effect on the onset f-I curve (the initial maximal firing rate following a step increase in an input current) measured at increasing mean threshold levels, i.e., adaptation states. In contrast, an adaptation current subtractively shifts this f-I curve to higher inputs without affecting its slope. As a consequence, an adaptation current acts essentially linearly, resulting in a high-pass filter component of the neuron's transfer function for current stimuli. With a dynamic threshold, however, the transfer function strongly depends on the input range because of the multiplicative effect on the f-I curves. Simulations of conductance-based spiking models with adaptation currents, such as afterhyperpolarization (AHP)-type, M-type, and sodium-activated potassium currents, do not show the divisive effects of a dynamic threshold, but agree with the properties of integrate-and-fire neurons with adaptation current. Notably, the effects of slow inactivation of sodium currents cannot be reproduced by either model. Our results suggest that, when lateral shifts of the onset f-I curve are seen in response to adapting inputs, adaptation should be modeled with adaptation currents and not with a dynamic threshold. In contrast, when the slope of onset f-I curves depends on the adaptation state, then adaptation should be modeled with a dynamic threshold. Further, the observation of divisively altered onset f-I curves in adapted neurons with notable variability of their spike threshold could hint to yet known biophysical mechanisms directly affecting the threshold.

Dynamics of deterministic and stochastic paired excitatory-inhibitory delayed feedback

We examine the effects of paired delayed excitatory and inhibitory feedback on a single integrate-and-fire neuron with reversal potentials embedded within a feedback network. These effects are studied using bifurcation theory and numerical analysis. The feedback occurs through modulation of the excitatory and inhibitory conductances by the previous firing history of the neuron; as a consequence, the feedback also modifies the membrane time constant. Such paired feedback is ubiquitous in the nervous system. We assume that the feedback dynamics are slower than the membrane time constant, which leads to a rate model formulation. Our article provides an extensive analysis of the possible dynamical behaviors of such simple yet realistic neural loops as a function of the balance between positive and negative feedback, with and without noise, and offers insight into the potential behaviors such loops can exhibit in response to time-varying external inputs. With excitatory feedback, the system can be quiescent, can be periodically firing, or can exhibit bistability between these two states. With inhibitory feedback, quiescence, oscillatory firing rates, and bistability between constant and oscillatory firing-rate solutions are possible. The general case of paired feedback exhibits a blend of the behaviors seen in the extreme cases and can produce chaotic firing. We further derive a condition for a dynamically balanced paired feedback in which there is neither bistability nor oscillations. We also show how a biophysically plausible smoothing of the firing function by noise can modify the existence and stability of fixed points and oscillations of the system. We take advantage in our mathematical analysis of the existence of an invariant manifold, which reduces the dimensionality of the dynamics, and prove the stability of this manifold. The novel computational challenges involved in analyzing such dynamics with and without noise are also described. Our results demonstrate that a paired delayed feedback loop can act as a sophisticated computational unit, capable of switching between a variety of behaviors depending on the input current, the relative strengths and asymmetry of the two parallel feedback pathways, and the delay distributions and noise level.

Coding of odor intensity

A model for coding of odor intensity in the first two neuronal layers of olfactory systems is proposed. First, the occupation and activation by odorant molecules of receptor proteins of different types borne by the first order neurons are described as birth and death processes. The occupation (birth) rate depends on the concentration of the odorant, whereas the probability of activation of an occupied receptor depends on the type of the odorant. Second, the spike generation mechanism proposed for the first order neuron depends on the level of the generator potential evoked by the activated receptors and on a time-decaying threshold which is reset to infinity after each spike. The various resulting stochastic regimes of firing activity at different concentrations are described. Third, each second order neuron is influenced by excitation coming from numerous first order neurons, lateral inhibition from other second order neurons, and self-inhibition. All these incoming signals are integrated at the second order neuron. The firing activity of the first and second order neurons is modeled by a first passage time scheme. For both types of neuron the shapes of the curves predicted by the model for the mean firing frequency as a function of stimulus concentration are shown to be in accordance with available experimental results.

On approximations of Stein's neuronal model

Stein's model represents a commonly-used description of spontaneous neuronal activity. Substituting Stein's model by the Ornstein-Uhlenbeck diffusion process increases the model tractability. A diffusion approximation of Stein's model is summarized in the present paper. It is proved that the cumulative distribution functions of interspike intervals under Stein's model converge to the cumulative distribution function of interspike intervals which are generated in accordance with the limiting Ornstein-Uhlenbeck diffusion model. The approach used allows us to determine to what extent Stein's model modifications and generalizations affect the possibility of diffusion approximation. It can be seen that non-diffusion approximations exist and they are also studied here. The results achieved can be considered as complementary to the numerical study published recently.

Excitation properties of the squid axon membrane and model systems with current stimulation. Statistical evaluation and comparison

The space-clamped squid axon membrane and two versions of the Hodgkin-Huxley model (the original, and a strongly adapting version) are subjected to a first order dynamic analysis. Stable, repetitive firing is induced by phase-locking nerve impulses to sinusoidal currents. The entrained impulses are then pulse position modulated by additional, small amplitude perturbation sinusoidal currents with respect to which the frequencies response of impulse density functions are measured. (Impulse density is defined as the number of impulses per unit time of an ensemble of membranes with each membrane subject to the same stimulus). Two categories of dynamic response are observed: one shows clear indications of a corner frequency, the other has the corner frequency obscured by dynamics associated with first order conductance perturbations in the interspike interval. The axon membrane responds with first order perturbations whereas the unmodified Hodgkin-Huxley model does not. Quantitative dynamic signatures suggest that the relaxation times of axonal recovery excitation variables are twice as long as those of the corresponding model variables. A number of other quantitative differences between axon and models, including the values of threshold stimuli are also observed.