Failure of Delayed Feedback Deep Brain Stimulation for Intermittent Pathological Synchronization in Parkinson's Disease

Suppression of excessively synchronous beta-band oscillatory activity in the brain is believed to suppress hypokinetic motor symptoms of Parkinson's disease. Recently, a lot of interest has been devoted to desynchronizing delayed feedback deep brain stimulation (DBS). This type of synchrony control was shown to destabilize the synchronized state in networks of simple model oscillators as well as in networks of coupled model neurons. However, the dynamics of the neural activity in Parkinson's disease exhibits complex intermittent synchronous patterns, far from the idealized synchronous dynamics used to study the delayed feedback stimulation. This study explores the action of delayed feedback stimulation on partially synchronized oscillatory dynamics, similar to what one observes experimentally in parkinsonian patients. We employ a model of the basal ganglia networks which reproduces experimentally observed fine temporal structure of the synchronous dynamics. When the parameters of our model are such that the synchrony is unphysiologically strong, the feedback exerts a desynchronizing action. However, when the network is tuned to reproduce the highly variable temporal patterns observed experimentally, the same kind of delayed feedback may actually increase the synchrony. As network parameters are changed from the range which produces complete synchrony to those favoring less synchronous dynamics, desynchronizing delayed feedback may gradually turn into synchronizing stimulation. This suggests that delayed feedback DBS in Parkinson's disease may boost rather than suppress synchronization and is unlikely to be clinically successful. The study also indicates that delayed feedback stimulation may not necessarily exhibit a desynchronization effect when acting on a physiologically realistic partially synchronous dynamics, and provides an example of how to estimate the stimulation effect.Comment: 19 pages, 8 figure

Exploring Neuronal Bistability at the Depolarization Block

Many neurons display bistability - coexistence of two firing modes such as bursting and tonic spiking or tonic spiking and silence. Bistability has been proposed to endow neurons with richer forms of information processing in general and to be involved in short-term memory in particular by allowing a brief signal to elicit long-lasting changes in firing. In this paper, we focus on bistability that allows for a choice between tonic spiking and depolarization block in a wide range of the depolarization levels. We consider the spike-producing currents in two neurons, models of which differ by the parameter values. Our dopaminergic neuron model displays bistability in a wide range of applied currents at the depolarization block. The Hodgkin-Huxley model of the squid giant axon shows no bistability. We varied parameter values for the model to analyze transitions between the two parameter sets. We show that bistability primarily characterizes the inactivation of the Na+ current. Our study suggests a connection between the amount of the Na+ window current and the length of the bistability range. For the dopaminergic neuron we hypothesize that bistability can be linked to a prolonged action of antipsychotic drugs.Comment: 26 pages, 8 figures, accepted to PLoS ON

Mathematical models of basal ganglia dynamics

Physical and biological phenomena that involve oscillations on multiple time scales attract attention of mathematicians because resulting equations include a small parameter that allows for decomposing a three- or higher-dimensional dynamical system into fast/slow subsystems of lower dimensionality and analyzing them independently using geometric singular perturbation theory and other techniques. However, in most life sciences applications observed dynamics is extremely complex, no small parameter exists and this approach fails. Nevertheless, it is still desirable to gain insight into behavior of these mathematical models using the only viable alternative – ad hoc computational analysis. Current dissertation is devoted to this latter approach. Neural networks in the region of the brain called basal ganglia (BG) are capable of producing rich activity patterns. For example, burst firing, i.e. a train of action potentials followed by a period of quiescence in neurons of the subthalamic nucleus (STN) in BG was shown to be related to involuntary shaking of limbs in Parkinson\u27s disease called tremor. The origin of tremor remains unknown; however, a few hypotheses of tremor-generation were proposed recently. The first project of this dissertation examines the BG-thalamo-cortical loop hypothesis for tremor generation by building physiologically-relevant mathematical model of tremor-related circuits with negative delayed feedback. The dynamics of the model is explored under variation of connection strength and delay parameters in the feedback loop using computational methods and data analysis techniques. The model is shown to qualitatively reproduce the transition from irregular physiological activity to pathological synchronous dynamics with varying parameters that are affected in Parkinson\u27s disease. Thus, the proposed model provides an explanation for the basal ganglia-thalamo-cortical loop mechanism of tremor generation. Besides tremor-related bursting activity BG structures in Parkinson\u27s disease also show increased synchronized activity in the beta-band (10-30Hz) that ultimately causes other parkinsonian symptoms like slowness of movement, rigidity etc. Suppression of excessively synchronous beta-band oscillatory activity is believed to suppress hypokinetic motor symptoms in Parkinson\u27s disease. Recently, a lot of interest has been devoted to desynchronizing delayed feedback deep brain stimulation (DBS). This type of synchrony control was shown to destabilize synchronized state in networks of simple model oscillators as well as in networks of coupled model neurons. However, the dynamics of the neural activity in Parkinson\u27s disease exhibits complex intermittent synchronous patterns, far from the idealized synchronized dynamics used to study the delayed feedback stimulation. The second project of this dissertation explores the action of delayed feedback stimulation on partially synchronous oscillatory dynamics, similar to what one observes experimentally in parkinsonian patients. We employ a computational model of the basal ganglia networks which reproduces the fine temporal structure of the synchronous dynamics observed experimentally. Modeling results suggest that delayed feedback DBS in Parkinson\u27s disease may boost rather than suppresses synchronization and is therefore unlikely to be clinically successful. Single neuron dynamics may also have important physiological meaning. For instance, bistability – coexistence of two stable solutions observed experimentally in many neurons is thought to be involved in some short-term memory tasks. Bistability that occurs at the depolarization block, i.e. a silent depolarized state a neuron enters with excessive excitatory input was proposed to play a role in improving robustness of oscillations in pacemaker-type neurons. The third project of this dissertation studies what parameters control bistability at the depolarization block in the three-dimensional conductance-based neuronal model by comparing the reduced dopaminergic neuron model to the Hodgkin-Huxley model of the squid giant axon. Bifurcation analysis and parameter variations revealed that bistability is mainly characterized by the inactivation of the Na+ current, while the activation characteristics of the Na+ and the delayed rectifier K+ currents do not account for the difference in bistability in the two models

On the origin of tremor in Parkinson's disease.

The exact origin of tremor in Parkinson's disease remains unknown. We explain why the existing data converge on the basal ganglia-thalamo-cortical loop as a tremor generator and consider a conductance-based model of subthalamo-pallidal circuits embedded into a simplified representation of the basal ganglia-thalamo-cortical circuit to investigate the dynamics of this loop. We show how variation of the strength of dopamine-modulated connections in the basal ganglia-thalamo-cortical loop (representing the decreasing dopamine level in Parkinson's disease) leads to the occurrence of tremor-like burst firing. These tremor-like oscillations are suppressed when the connections are modulated back to represent a higher dopamine level (as it would be the case in dopaminergic therapy), as well as when the basal ganglia-thalamo-cortical loop is broken (as would be the case for ablative anti-parkinsonian surgeries). Thus, the proposed model provides an explanation for the basal ganglia-thalamo-cortical loop mechanism of tremor generation. The strengthening of the loop leads to tremor oscillations, while the weakening or disconnection of the loop suppresses them. The loop origin of parkinsonian tremor also suggests that new tremor-suppression therapies may have anatomical targets in different cortical and subcortical areas as long as they are within the basal ganglia-thalamo-cortical loop

Simulated effect of antipsychotic drugs in the DA neuron.

<p>The influence of the drugs is modeled as excitation by applied current (lower trace) Tonic firing in the DA neuron (upper trace) is interrupted with excessive excitation. DA neuron remains silent after complete withdrawal of excitation due to hysteresis. Parameters for the DA neuron are from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0042811#pone-0042811-t001" target="_blank">Table 1</a>.</p

Two-parameter bifurcation diagrams of the DA and HH neuron for the change in slope factors of (in)activation functions.

<p>The hysteresis regions are shaded gray. A) A gradual voltage dependence of the Na⁺ current inactivation function removes hysteresis in the DA neuron. B) Steeper voltage dependence of the activation of the Na⁺ current has almost no effect on hysteresis in the DA neuron. C) More gradual voltage dependence of the K⁺ current has little effect on hysteresis in the DA neuron. Parameters are from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0042811#pone-0042811-g004" target="_blank">Figure 4B</a>. D) More gradual voltage dependence of the Na⁺ current reduces and then completely abolishes hysteresis in the HH neuron. E) Gradual voltage dependence of the activation of the Na⁺ current has little effect on hysteresis at the upper boundary of oscillatory region in the HH neuron. F) Hysteresis range peaks at intermediate values of <i>S</i>_n in the HH neuron. A solid curve represents an Andronov-Hopf bifurcation, a dashed curve – a saddle-node bifurcation of limit cycles. Horizontal dotted lines in A) and D) mark the value of slope parameter from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0042811#pone-0042811-t001" target="_blank">Table 1</a> for the HH neuron.</p

Parameter values for STN and GPe neurons.

<p>Parameter values for STN and GPe neurons.</p

The K⁺ current dynamics in the DA and HH neurons.

<p>A) Activation and B) time constant functions of the K⁺ current from the DA neuron (solid curve) and the HH neuron (dashed curve). The ranges for the HH neuron are at the top and to the right.</p

Changing kinetics of gating variables in the DA neuron and the HH neuron.

<p>The hysteresis regions are shaded gray. A), B) Bistability range shortens with accelerating the gating variables kinetics in the DA neuron. <i>f</i>_h = 1 and <i>f</i>_n = 1 correspond to parameter set from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0042811#pone-0042811-g004" target="_blank">Figure 4B</a>. C) Simultaneous acceleration of both n and h variables decreases the size of bistability range. <i>f</i>_n,h = 1 corresponds to parameter set from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0042811#pone-0042811-g004" target="_blank">Figure 4B</a>. D), E) Hysteresis is reduced or eliminated in the HH neuron with slowing the individual current kinetics. <i>f</i>_h = 1 and <i>f</i>_n = 1 correspond to parameter set from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0042811#pone-0042811-g004" target="_blank">Figure 4D</a>. F) Hysteresis is increased with simultaneous slowing of gating variables <i>n</i> and <i>h</i>. <i>f</i>_n,h = 1 corresponds to parameter set from <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0042811#pone-0042811-g004" target="_blank">Figure 4D</a>. A solid curve represents an Andronov-Hopf bifurcation, a dashed curve – a saddle-node bifurcation of limit cycles. Horizontal dotted lines (where shown) give the values of <i>f</i>_h and <i>f</i>_n for which the maximum value of the corresponding time constant function for the DA (HH) neuron matches the maximum value of the time constant for the HH (DA) neuron.</p

Tremulous activity with variation of dopaminergic parameters.

<p>The parameters <i>s</i>₁ and <i>s</i>₂ run along vertical and horizontal axis respectively, color codes for the value of SNR. The point (1, 1) corresponds to the bursting mode shown in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0041598#pone-0041598-g002" target="_blank">Figure 2B</a>. A), B), C), and D) represent SNR1, 2, 3, and 4 respectively. Parameters <i>s</i>₁ and <i>s</i>₂ are proxies of dopaminergic status and their higher values correspond to stronger dopamine influence. Thus upper right corner corresponds to a “normal” state of the network, while lower left corner corresponds to a “parkinsonian” state. Blue color indicates the absence of tremor-band oscillations, red color indicates prominent oscillations. Yellow and green correspond to SNR values termed to be tremulous in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0041598#pone.0041598-Ramanathan1" target="_blank">[48]</a>.</p