Mathematical Sciences
http://hdl.handle.net/10468/88
Thu, 31 Aug 2017 20:02:14 GMT2017-08-31T20:02:14ZModeling cortical spreading depression induced by the hyperactivity of interneurons
http://hdl.handle.net/10468/4018
Modeling cortical spreading depression induced by the hyperactivity of interneurons
Desroches, Mathieu; Faugeras, Olivier; Krupa, Martin; Mantegazza, Massimo
Cortical spreading depression (CSD) is a wave of transient intense neuronal firing leading to a long lasting depolarization block of neuronal activity. It is a proposed pathological mechanism of migraine with aura. Some molecular/cellular mechanisms of migraine with aura and of CSD have been identified studying a rare mendelian form: familial hemiplegic migraine (FHM). FHM type 1 & 2 are caused by mutations of the CaV2.1 Ca2+ channel and the glial Na+ / K+ pump, respectively, leading to facilitation of CSD in mouse models mainly because of increased glutamatergic transmission/extracellular glutamate build-up. FHM type 3 mutations of the SCN1A gene, coding for the voltage gated sodium channel NaV1.1, cause gain of function of the channel and hyperexcitability of GABAergic interneurons. This leads to the counterintuitive hypothesis that intense firing of interneurons can cause CSD ignition. To test this hypothesis in silico, we developed a computational model of an E-I pair (a pyramidal cell and an interneuron), in which the coupling between the cells in not just synaptic, but takes into account also the effects of the accumulation of extracellular potassium caused by the activity of the neurons and of the synapses. In the context of this model, we show that the intense firing of the interneuron can lead to CSD. We have investigated the effect of various biophysical parameters on the transition to CSD, including the levels of glutamate or GABA, frequency of the interneuron firing and the efficacy of the KCC2 co-transporter. The key element for CSD ignition in our model was the frequency of interneuron firing and the related accumulation of extracellular potassium, which induced a depolarization block of the pyramidal cell. Our model can be used to study other types of activities in microcircuits and of couplings between excitatory and inhibitory neurons.
Wed, 10 May 2017 00:00:00 GMThttp://hdl.handle.net/10468/40182017-05-10T00:00:00ZNonlinearization and waves in bounded media: old wine in a new bottle
http://hdl.handle.net/10468/3772
Nonlinearization and waves in bounded media: old wine in a new bottle
Mortell, Michael P.; Seymour, Brian R.
We consider problems such as a standing wave in a closed straight tube, a self-sustained oscillation, damped resonance, evolution of resonance and resonance between concentric spheres. These nonlinear problems, and other similar ones, have been solved by a variety of techniques when it is seen that linear theory fails. The unifying approach given here is to initially set up the appropriate linear difference equation, where the difference is the linear travel time. When the linear travel time is replaced by a corrected nonlinear travel time, the nonlinear difference equation yields the required solution.
Wed, 01 Mar 2017 00:00:00 GMThttp://hdl.handle.net/10468/37722017-03-01T00:00:00ZExploring diurnal variation using piecewise linear splines: an example using blood pressure
http://hdl.handle.net/10468/3752
Exploring diurnal variation using piecewise linear splines: an example using blood pressure
Madden, Jamie M.; Li, Xia; Kearney, Patricia M.; Tilling, Kate; Fitzgerald, Anthony P.
Background: There are many examples of physiological processes that follow a circadian cycle and researchers are interested in alternative methods to illustrate and quantify this diurnal variation. Circadian blood pressure (BP) deserves additional attention given uncertainty relating to the prognostic significance of BP variability in relation to cardiovascular disease. However, the majority of studies exploring variability in ambulatory blood pressure monitoring (ABPM) collapse the data into single readings ignoring the temporal nature of the data. Advanced statistical techniques are required to explore complete variation over 24 h. Methods: We use piecewise linear splines in a mixed-effects model with a constraint to ensure periodicity as a novel application for modelling daily blood pressure. Data from the Mitchelstown Study, a cross-sectional study of Irish adults aged 47–73 years (n = 2047) was utilized. A subsample (1207) underwent 24-h ABPM. We compared patterns between those with and without evidence of subclinical target organ damage (microalbuminuria). Results: We were able to quantify the steepest rise and fall in SBP, which occurred just after waking (2.23 mmHg/30 min) and immediately after falling asleep (−1.93 mmHg/30 min) respectively. The variation about an individual’s trajectory over 24 h was 12.3 mmHg (standard deviation). On average those with microalbuminuria were found to have significantly higher SBP (7.6 mmHg, 95% CI 5.0–10.1) after adjustment for age, sex and BMI. Including an interaction term between each linear spline and microalbuminuria did not improve model fit. Conclusion: We have introduced a practical method for the analysis of ABPM where we can determine the rate of increase or decrease for different periods of the day. This may be particularly useful in examining chronotherapy effects of antihypertensive medication. It offers new measures of short-term BP variability as we can quantify the variation about an individual’s trajectory but also allows examination of the variation in slopes between individuals (random-effects).
Thu, 02 Feb 2017 00:00:00 GMThttp://hdl.handle.net/10468/37522017-02-02T00:00:00ZRandom walks on finite quantum groups
http://hdl.handle.net/10468/4033
Random walks on finite quantum groups
McCarthy, J.P.
Of central interest in the study of random walks on finite groups are ergodic random walks. Ergodic random walks converge to random in the sense that as the number of transitions grows to infinity, the state-distribution converges to the uniform distribution on G. The study of random walks on finite groups is generalised to the study of random walks on quantum groups. Quantum groups are neither groups nor sets and rather what are studied are finite dimensional algebras that have the same properties as the algebra of functions on an actual group — except for commutativity. The concept of a random walk converging to random — and a metric for measuring the distance to random after k transitions — is generalised from the classical case to the case of random walks on quantum groups. A central tool in the study of ergodic random walks on finite groups is the Upper Bound Lemma of Diaconis and Shahshahani. The Upper Bound Lemma uses the representation theory of the group to generate upper bounds for the distance to random and thus can be used to determine convergence rates for ergodic walks. The representation theory of quantum groups is very well understood and is remarkably similar to the representation theory of classical groups. This allows for a generalisation of the Upper Bound Lemma to an Upper Bound Lemma for quantum groups. The Quantum Diaconis–Shahshahani Upper Bound Lemma is used to study the convergence of ergodic random walks on classical groups Zn, Z n 2 , the dual group Scn as well as the ‘truly’ quantum groups of Kac and Paljutkin and Sekine. Note that for all of these generalisations, restricting to commutative subalgebras gives the same definitions and results as the classical theory.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10468/40332017-01-01T00:00:00Z