Dependencia e Improductividad como forma de Asistencialismo

Hopf bifurcation. The green curve shows the border where the transition from bursting into silent regimes occurs (Cymbalyuk, G.S., et al., 2002). The red dots mark the points where the homoclinic bifurcation occurs.

5 10 15 20 −70 −65 −60 −55 −50 g leak (nS) E leak (mV)

We have shown that the unstable hyperpolarized sub-threshold oscillations (periodic orbit) emerge through the Andronov-Hopf bifurcation and disappear through the homoclinic bifurcation. Diagram 3 shows the Andronov-Hopf bifurcation curve on which the second border of the area of the co-existence is defined.

In conclusion, we showed that the unstable sub-threshold oscillations separate the stable equilibrium (rest state) and bursting activity. Minimum and maximum values of the voltage of the unstable periodic orbit define two thresholds: upper and lower. It explained the series of numerical experiments where the switch between different regimes can be produced by a pulse of current of certain amplitude. We showed the area of co-existence of bursting activity and silence is defined by the sub-critical Andronov-Hopf bifurcation and a homoclinic bifurcation. The homoclinic bifurcation explains the transition from bursting into silence.

DISCUSSION

Basic patterns of neuronal activities are silence, sub-threshold oscillations, tonic spiking and bursting activity. A number of theoretical and experimental studies have demonstrated that both a single neuron and neuronal populations can exhibit bi-stability, in other words, these regimes can co-exist with one another (Wang et al., 1995; Shilnikov et al., 2004; Milton and Jung, 2003; Cymbalyuk et al., 2002; Cymbalyuk et al., 2000; Feoden and Grebogi, 1997; Gadatela and Dangelmayr, 2001; Manning et al., 2003; Manuca et al., 1998).

In cortical neurons, many classes display transitions between tonic spiking and bursting as a function of the brain state, for example sleep versus wakefulness (Steriade, 2001). The co- existence of tonic firing and bursting has been shown in cortical neurons (Fröhlich. and Bazhenov, 2006). Depolarization of a neuron caused by raised extracellular K+ concentration can lead to bursting in a cell which usually shows tonic spiking. (Jensen et al., 1997). The increase of K+ concentration during epileptogenesis is an established fact (Somjen et al., 2004; DeLorenzo et al., 2005). Frohlich and Bazhenov developed a cortical neuron model which explains bi-stability between tonic spiking and bursting for raised K+ _{concentration. The leech heart interneuron}

model also exhibits co-existence of two types of oscillations: tonic spiking and bursting with large amplitude (Shilnikov et al., 2004). Switching between these two regimes can be produced by modulation of the initial state of the neuron model (Shilnikov et al., 2004). Bursting is separated from tonic spiking by an unstable periodic tonic spiking activity of the saddle type, which determines the threshold between regimes.

Here, we investigated a mechanism of bi-stability of the model of the leech heart interneuron. We have illustrated its properties by a series of numerical experiments. They display

that the activity of the cell which was initially in the silent regime can be switched into bursting activity by depolarized and hyperpolarized pulses of the current. In the other series of experiments, where the neuron was initially in the bursting mode, we showed that it can be switched into the silent mode by a hyperpolarized pulse. In the later case, if the amplitude of the hyperpolarized pulse is increased, the cell returns to bursting activity. These numerical experiments imply two thresholds: upper and lower. We have shown that switching between bursting activity and silence can be controlled by a pulse of current.

The knowledge of the mechanism of bi-stability is useful in the design of treatments of such medical conditions as Parkinson’s disease, sudden infant death syndrome (SIDS), epilepsy, and essential tremor (Breakspear et al., 2006; Harvey et al., 2007; Milton and Jung, 2003; Wang et al., 1995; DeLorenzo, 2005; Weese-Mayer et al., 2007). What is common for these medical conditions is that they are examples of dynamically dysfunctional activities of neuronal system

(Feudel et al., 1997; Gadaleta et al., 2001; Manning et al., 2003; Milton and Jung, 2003). For example, SIDS is defined as a sudden, unexpected death of an apparently healthy infant under one year of age, usually during sleep (Weese-Mayer et al., 2007). It has been suggested that this syndrome could be a result of the bi-stability of the neuronal system, where death is associated with the silent regime.

Damage of the neurons might cause a change in the activity of the whole neural network and lead to epilepsy (DeLorenzo et al., 2005; Milton and Jung, 2003). Epilepsy can be generated in a large group of cells. It is the most common serious condition in neurology. Seizures are frequently accompanied with synchronized hyperactivity in the neuronal network and the cells are depolarized to a high level (Drongelen et al., 2003). A weak electrical field can modulate neuronal activity (Francis et al., 2003). It has been shown that deep brain stimulation can provide

noticeable benefits for people with tremor or Parkinson’s disease. High frequency stimulation of the intermediate nucleus of the thalamus essentially could relieve tremors (Perlmutter et al., 2006). There are a number of experiments in which the seizure can be stopped by a pulse of current (Milton and Jung, 2003; Perlmutter et al., 2006).

Exemplary ideas of bi-stability are exploited in designs of treatment of Parkinson’s disease. Under normal conditions, particular neuronal populations located in the thalamus and the basal ganglia show apparently random electrical activity, whereas abnormal synchronized activity of these neurons is associated with Parkinson’s resting tremors (Alberts et al., 1969). The main question is how to suppress that abnormal synchronized activity. Tass and Hauptmann created a model of the subthalamic nucleus (STN) (Tass and Hauptmann, 2007). Its analysis shows that appropriate electrical stimulation of the bursting neurons located in the thalamus may change synaptic wiring of the neuronal networks. It gives rise to the co-existence of different dynamically stable regimes. Due to this multistability of the neuronal network, it is possible to switch the network activity from the strongly synchronized state to a desynchronized state. It has been proposed that suppression of synchronized activity is based on two plasticity induced effects. The first is desynchronizing stimulation, which causes a decrease of synaptical weights. The second effect requires that the weakly synchronized state must be stable (Tass, 2001). To solve the problem of abnormal synchronization, authors have suggested applying weak or short stimulation. This stimulation causes a reduction of a pathologically synchronized neuronal population (Tass, 2001; Tass and Hauptmann, 2007). The main advantage of such stimulation, that it does not affect memory as much as alternative methods which are based upon longer stimulation. To induce the switch from abnormal to normal neuronal activity, Tass suggests using a two pulses deep brain stimulation techniques.

Discussing treatment of epileptic seizures, Milton brought into consideration the co- existence of two patterns of tonic spiking activities that arise in membrane potential of an Aplisia

motoneuron. When the single stimulation is applied the switch between activities occurs (Milton and Jung, 2003). Designing methods of feedback stimulation, Milton shows that multistability can appear in the models of delayed feedback control. The necessary conditions for multistability were established as following; the sum of the conduction times along the axon and dendrites, time required for quantal release, processing times in the neurons, and the rise time of synaptic inhibition has to be greater than the intrinsic firing period of the neuron. Multistability was presented as the multiple basins of attraction separated by ridges of varying heights that correspond to energy barriers (Milton and Jung, 2003). Such a dynamical system can be controlled and manipulated with small perturbation. Milton has designed a feedback stimulation method of regulation of neuronal network activity. At each moment of time when the abnormal attractor occurs, the feedback stimulation imposes the initial conditions for the normal attractor (Milton and Jung, 2003 ).

In 2002, Netoff and Schiff have proposed that a decrease in synchronization is essential for initiation and maintenance of epileptic seizures. If synchrony is associated with seizure termination, then the method directed at increasing such synchronization may be useful in controlling seizures (Netoff and Schiff, 2002).

Our results explain a novel mechanism bi-stability at the cellular level. If neurons in a population where each neuron possess the property of bi-stability of bursting and silence according to the mechanism described here, then we can provide specific instructions for the design of electrical stimulation. According to the scenario described, two attractors are separated by the unstable periodic orbit, which was born through Andronov-Hopf bifurcation. The

perturbation by the pulse of current can be applied to the group of cells. If the pulse of current is injected with appropriate amplitude and at the specific phase, the bursting can be locked into the silent regime. The theory of bi-stability and the controlling mechanism might be applied to the design of a treatment of tissue displaying epileptic behavior. The feedback techniques may suppress the abnormal activity of the neuron by switching into the normal regime.

In conclusion, we described a novel mechanism explaining the co-existence of bursting and silent regimes in a single neuron. We demonstrated how the injection current pulse can be applied with the certain amplitude and at the specific phase to make a switch between regimes. We demonstrated that the specific characteristics of the pulse are defined by the unstable sub- threshold oscillations which separate two basins of attraction (bursting and silence). The mechanism of supporting the co-existence is similar for both the simplified four dimensional model and the complete fourteen dimensional model. In both models the area of co-existence is determined by two bifurcations: Homoclinic and Andronov-Hopf. The hypothesis is that the area of the co-existence of bursting and silence is defined by two types of bifurcations, which might give a general principal for controlling bi-stability in a Hodgkin-Huxley type model.

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