Fold cobordisms and a Poincare-Hopf type theorem for the signature

We give complete geometric invariants of cobordisms of framed fold maps. These invariants consist of two types. We take the immersion of the fold singular set into the target manifold together with information about non-triviality of the normal bundle of the singular set in the source manifold. These invariants were introduced in the author's earlier works. Secondly we take the induced stable partial framing on the source manifold whose cobordisms were studied in general by Koschorke. We show that these invariants describe completely the cobordism groups of framed fold maps into R^n. Then we are looking for dependencies between these invariants and we study fold maps of 4k-dimensional manifolds into R^2. We construct special fold maps which are representatives of the fold cobordism classes and we also compute cobordism groups. We obtain a Poincare-Hopf type formula, which connects local data of the singularities of a fold map of an oriented 4k-dimensional manifold M to the signature of M. We also study the unoriented case analogously and prove a similar formula about the twisting of the normal bundle of the fold singular set.Comment: 39 pages, 3 figures, revise

Stability of a chain of phase oscillators

We study a chain of N + 1 phase oscillators with asymmetric but uniform coupling. This type of chain possesses 2 N ways to synchronize in so-called traveling wave states, i.e., states where the phases of the single oscillators are in relative equilibrium. We show that the number of unstable dimensions of a traveling wave equals the number of oscillators with relative phase close to π . This implies that only the relative equilibrium corresponding to approximate in-phase synchronization is locally stable. Despite the presence of a Lyapunov-type functional, periodic or chaotic phase slipping occurs. For chains of lengths 3 and 4 we locate the region in parameter space where rotations (corresponding to phase slipping) are present

Estimates of persistent inward current in human motor neurons during postural sway

Persistent inward current (PIC) is a membrane property critical for increasing gain of motor neuron output. In humans, most estimates of PIC are made from plantarflexor or dorsiflexor motor units with the participant in a seated position with the knee flexed. This seated and static posture neglects the task-dependent nature of the monoaminergic drive that modulates PIC activation. Seated estimates may drastically underestimate the amount of PIC that occurs in human motor neurons during functional movement. The current study estimated PIC using the conventional paired motor unit technique which uses the difference between reference unit firing frequency at test unit recruitment and reference unit firing frequency at test unit de-recruitment (∆F) during triangular-shaped, isometric ramps in plantarflexion force as an estimate of PIC. Estimates of PIC were also made during standing anterior postural sway, a postural task that elicits a ramped increase and decrease in soleus motor unit activation similar to the conventional seated ramp contractions. For each motor unit pair, ∆F estimates of PIC made during conventional isometric ramps in the seated posture were compared to those made during standing postural sway. Baseline reciprocal inhibition (RI) was also measured in each posture using the post-stimulus time histogram (PSTH) technique. Hyperpolarizing input has been shown to have a reciprocal relationship with PIC in seated posture and RI was measured to examine if the same reciprocal relationship holds true during functional PIC estimation. It was hypothesized that an increase in ∆F would be seen during standing compared to sitting due to greater neuromodulatory input. We found that ∆F estimates during standing postural sway were equal (2.44 ± 1.17, p=0.44) to those in seated PIC estimates (2.73± 1.20) using the same motor unit pair. Reciprocal inhibition was significantly lower when measured in a standing posture (0.0031 ± 0.0251,