Mode locking in a periodically forced "ghostbursting" neuron model

We study a minimal integrate-and-fire based model of a "ghostbursting" neuron under periodic stimulation. These neurons are involved in sensory processing in weakly electric fish. There exist regions in parameter space in which the model neuron is mode-locked to the stimulation. We analyse this locked behavior and examine the bifurcations that define the boundaries of these regions. Due to the discontinuous nature of the flow, some of these bifurcations are nonsmooth. This exact analysis is in excellent agreement with numerical simulations, and can be used to understand the response of such a model neuron to biologically realistic input.

Toxicity of 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD) in the developing male Wistar(Han) rat I: no decrease in epididymal sperm count after a single acute dose

It has been reported that fetal exposure to 2,3,7,8-tetrachlorodibenzo-p-dioxin (TCDD) causes defects in the male reproductive system of the rat. We set out to replicate and extend these effects using a robust experimental design. Groups of 75 (control vehicle) or 55 (50, 200 or 1000 ng of TCDD kg-1 bodyweight) female Wistar(Han) rats were exposed to TCDD on Gestational Day (GD) 15, then allowed to litter. The high dose group dams showed no sustained weight loss compared to control, but four animals had total litter loss. Pups in the high dose group showed reduced body weight up till day 21, and pups in the medium dose group showed reduced body weight in the first week post partum. Balano-preputial separation (BPS) was significantly delayed in the high dose group male offspring. There were no significant effects of treatment when the offspring were subjected to a functional observational battery, or mated with females to assess reproductive capability. 25 males per group were killed on post natal day (PND) 70, and ~60 animals per group (~30 for the high dose group) on PND120 to assess seminology and other endpoints. At PND120, the two highest dose groups showed a statistically significant elevation of sperm counts, compared to control; however, this effect was small (~30%), within the normal range of sperm counts for this strain of rat, was not reflected in testicular spermatid counts nor PND70 data, and is therefore postulated to have no biological significance. Although there was an increase in the proportion of abnormal sperm at PND70, seminology parameters were otherwise unremarkable. Testis weights in the high dose group were slightly decreased at PND 70 and 120, and at PND120, brain weights were decreased in the high dose group, liver to body weight ratios were increased for all three dose groups, with an increase in inflammatory cell foci in the epididymis in the high dose group. These data show that TCDD is a potent developmental toxin after exposure of the developing fetus, but that acute developmental exposure to TCDD on GD15 caused no decrease in sperm counts.

Neuronal networks with gap junctions: A study of piece-wise linear planar neuron models

The presence of gap junction coupling among neurons of the central nervous systems has been appreciated for some time now. In recent years there has been an upsurge of interest from the mathematical community in understanding the contribution of these direct electrical connections between cells to large-scale brain rhythms. Here we analyze a class of exactly soluble single neuron models, capable of producing realistic action potential shapes, that can be used as the basis for understanding dynamics at the network level. This work focuses on planar piece-wise linear models that can mimic the firing response of several different cell types. Under constant current injection the periodic response and phase response curve (PRC) is calculated in closed form. A simple formula for the stability of a periodic orbit is found using Floquet theory. From the calculated PRC and the periodic orbit a phase interaction function is constructed that allows the investigation of phase-locked network states using the theory of weakly coupled oscillators. For large networks with global gap junction connectivity we develop a theory of strong coupling instabilities of the homogeneous, synchronous and splay state. For a piece-wise linear caricature of the Morris-Lecar model, with oscillations arising from a homoclinic bifurcation, we show that large amplitude oscillations in the mean membrane potential are organized around such unstable orbits.

Mode locking in a spatially extended neuron model: active soma and compartmental tree

Understanding the mode-locked response of excitable systems to periodic forcing has important applications in neuroscience. For example it is known that spatially extended place cells in the hippocampus are driven by the theta rhythm to generate a code conveying information about spatial location. Thus it is important to explore the role of neuronal dendrites in generating the response to periodic current injection. In this paper we pursue this using a compartmental model, with linear dynamics for each compartment, coupled to an active soma model that generates action potentials. By working with the piece-wise linear McKean model for the soma we show how the response of the whole neuron model (soma and dendrites) can be written in closed form. We exploit this to construct a stroboscopic map describing the response of the spatially extended model to periodic forcing. A linear stability analysis of this map, together with a careful treatment of the non-differentiability of the soma model, allows us to construct the Arnol'd tongue structure for 1:q states (one action potential for q cycles of forcing). Importantly we show how the presence of quasi-active membrane in the dendrites can influence the shape of tongues. Direct numerical simulations confirm our theory and further indicate that resonant dendritic membrane can enlarge the windows in parameter space for chaotic behavior. These simulations also show that the spatially extended neuron model responds differently to global as opposed to point forcing. In the former case spatio-temporal patterns of activity within an Arnol'd tongue are standing waves, whilst in the latter they are traveling waves.