Deterministic and stochastic dynamics of multi-variable neuron models : resonance, filtered fluctuations and sodium-current inactivation

Neurons are the basic elements of the networks that constitute the computational units of the brain. They dynamically transform input information into sequences of electrical pulses. To conceive the complex function of the brain, it is crucial to understand this transformation and identify simple neuron models which accurately reproduce the known features of biological neurons. This thesis addresses three different features of neurons. We start by exploring the effect of subthreshold resonance on the response of a periodically forced neuron using a simple threshold model. The response is studied in terms of an implicit one-dimensional time map that corresponds to the Poincar´e map of the forced system. Qualitatively distinct responses are found, including mode locking and chaos. We analytically find the stability regions of mode-locking solutions, and identify the transition to chaos through period-adding bifurcations. We show that the response becomes chaotic when the forcing frequency is close to the resonant frequency. Then we will consider an experimentally verified model with realistic spikegenerating mechanism and study the effect of filtered synaptic fluctuations on the firing-rate response of the neuron. Using a population density method as well as an efficient numerical method, we find the steady-state firing rate in two limits of fast and slow synaptic inputs and present the linear response theory for the firing rate of the model in response to both time-dependent mean inputs and time-dependent noise intensity. Finally, a novel model is introduced that incorporates threshold variability of neurons. We determine the modulation of the input-output properties of the model due to oscillatory inputs and in the presence of filtered synaptic fluctuations.EThOS - Electronic Theses Online ServiceUniversity of WarwickOverseas Research Students Awards Scheme (ORSAS)GBUnited Kingdo

Scale-Free Navigational Planning by Neuronal Traveling Waves

Spatial navigation and planning is assumed to involve a cognitive map for evaluating trajectories towards a goal. How such a map is realized in neuronal terms, however, remains elusive. Here we describe a simple and noise-robust neuronal implementation of a path finding algorithm in complex environments. We consider a neuronal map of the environment that supports a traveling wave spreading out from the goal location opposite to direction of the physical movement. At each position of the map, the smallest firing phase between adjacent neurons indicate the shortest direction towards the goal. In contrast to diffusion or single-wave-fronts, local phase differences build up in time at arbitrary distances from the goal, providing a minimal and robust directional information throughout the map. The time needed to reach the steady state represents an estimate of an agent's waiting time before it heads off to the goal. Given typical waiting times we estimate the minimal number of neurons involved in the cognitive map. In the context of the planning model, forward and backward spread of neuronal activity, oscillatory waves, and phase precession get a functional interpretation, allowing for speculations about the biological counterpart

Deterministic and stochastic dynamics of multi-variable neuron models : resonance, filtered fluctuations and sodium-current inactivation

Neurons are the basic elements of the networks that constitute the computational units of the brain. They dynamically transform input information into sequences of electrical pulses. To conceive the complex function of the brain, it is crucial to understand this transformation and identify simple neuron models which accurately reproduce the known features of biological neurons. This thesis addresses three different features of neurons. We start by exploring the effect of subthreshold resonance on the response of a periodically forced neuron using a simple threshold model. The response is studied in terms of an implicit one-dimensional time map that corresponds to the Poincar´e map of the forced system. Qualitatively distinct responses are found, including mode locking and chaos. We analytically find the stability regions of mode-locking solutions, and identify the transition to chaos through period-adding bifurcations. We show that the response becomes chaotic when the forcing frequency is close to the resonant frequency. Then we will consider an experimentally verified model with realistic spikegenerating mechanism and study the effect of filtered synaptic fluctuations on the firing-rate response of the neuron. Using a population density method as well as an efficient numerical method, we find the steady-state firing rate in two limits of fast and slow synaptic inputs and present the linear response theory for the firing rate of the model in response to both time-dependent mean inputs and time-dependent noise intensity. Finally, a novel model is introduced that incorporates threshold variability of neurons. We determine the modulation of the input-output properties of the model due to oscillatory inputs and in the presence of filtered synaptic fluctuations

Scale-Free Navigational Planning by Neuronal Traveling Waves

<div><p>Spatial navigation and planning is assumed to involve a cognitive map for evaluating trajectories towards a goal. How such a map is realized in neuronal terms, however, remains elusive. Here we describe a simple and noise-robust neuronal implementation of a path finding algorithm in complex environments. We consider a neuronal map of the environment that supports a traveling wave spreading out from the goal location opposite to direction of the physical movement. At each position of the map, the smallest firing phase between adjacent neurons indicate the shortest direction towards the goal. In contrast to diffusion or single-wave-fronts, local phase differences build up in time at arbitrary distances from the goal, providing a minimal and robust directional information throughout the map. The time needed to reach the steady state represents an estimate of an agent’s waiting time before it heads off to the goal. Given typical waiting times we estimate the minimal number of neurons involved in the cognitive map. In the context of the planning model, forward and backward spread of neuronal activity, oscillatory waves, and phase precession get a functional interpretation, allowing for speculations about the biological counterpart.</p></div

Results in a network of 10 × 10 planning neurons with different noise scenarios.

<p>Start position at (7,9), goal position at (7,3) and obstacle indicated by the black bar. Top row: voltage of the four nearest neighbor neurons at the start position (identity color coded). Middle row: external input current these neurons receive. Bottom row: 2D environment with the chosen path (blue line) from the start to the goal. Standard deviations <i>σ</i>_<i>ext</i> of the input currents indicated below. a, b: The mean of the input current generated by the background Poisson spiking neuron was constant and identical for all neurons (<i>μ</i>_<i>ext</i> = 12 mV/ms), except for the goal neuron (marked with <i>g</i>, </p><p></p><p></p><p></p><p><mi>μ</mi></p><p><mi>e</mi><mi>x</mi><mi>t</mi></p><p><mi>g</mi><mi>o</mi><mi>a</mi><mi>l</mi></p><p></p><mo>=</mo><mn>12</mn><mo>.</mo><mn>5</mn><p></p><p></p><p></p>). c: A common sinusoidal fluctuation in the Poisson firing rate of the background neurons does not disturb the relative timing among neighboring neurons. d: A randomly chosen bias in the mean <p></p><p></p><p></p><p><mi>μ</mi></p><p><mi>e</mi><mi>x</mi><mi>t</mi></p><p><mi>i</mi><mi>j</mi></p><p></p><p></p><p></p><p></p> of the individual input currents with standard deviation 3 does not prevent the agent from finding a short path to the goal. In all simulations, the planning times were 600 ms, the readout times 250 ms, and the coupling strength was <i>ϵ</i> = 0.15.<p></p

Network architecture, traveling waves and planning time.

<p>(a) Planning and readout network. For each neuron in the planning layer, 4 actions can be assigned. Actions neurons associated to planning neuron (<i>i</i>, <i>j</i>) are <i>W</i>_<i>ij</i>, <i>E</i>_<i>ij</i>, <i>N</i>_<i>ij</i> and <i>S</i>_<i>ij</i> which receive synaptic input respectively from the left, right, north, and south neighbor of the neuron (<i>i</i>, <i>j</i>) and evoke a motion in the same directions (just one synaptic input is shown). Action neurons corresponding to the current place of the agent (here again (<i>i</i>, <i>j</i>)) are driven by an additional input (</p><p></p><p></p><p></p><p><mi>I</mi></p><p><mi>e</mi><mi>x</mi><mi>t</mi></p><p><mi>a</mi></p><p><mi>i</mi><mi>j</mi></p><p></p><p></p><p></p><p></p><p></p>). The first of the 4 action neurons that is fired by the passing traveling wave inhibits the other 3 action neurons. (b) Synaptically propagating waves of activity from the goal neuron at (1,1) across a planning layer of 20 × 20 neurons, for four different obstacle configurations. Colors code for firing phases at steady state relative to the goal neuron. (c) Time courses of the local phase difference for two sample neurons at positions (10,10) and (20,20), indicated by □ and ○ in the top left panel of b, with their local west-positioned neighbors. The time to reach the maximal local phase difference represents the planning time for these two start positions towards the goal (here 600 and 1050 ms, vertical lines), and subsequently the full path towards the goal can be read out.<p></p

Effect of noise on planning performance and readout time in the network used in Fig 3.

<p>(a) Planning performance, shown for 3 different readout times of 60, 180 and 240 ms (corresponding to 1, 3 and 4 readout cycles, bottom to top), declines with increasing noise (average across 10 chosen paths, error bars represent standard deviations of mean). (b) Readout time used at each position such that a shortest 10-step path is found, evaluated for the different noise levels. Parameters, network- and task configuration as used in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0127269#pone.0127269.g003" target="_blank">Fig 3</a>.</p

Determinants of the planning time.

<p>(a) Snapshots of the local phase differences along diagonal positions from the goal. The intrinsic oscillation frequency was 17 Hz for the non-goal positions and 18 Hz for the goal position. Noise level was <i>σ</i>_<i>ext</i> = 0. (b) Planning time (i.e. the time to reach roughly 90% of the final local phase difference) increases linearly with the distance from the goal. (c) Planning time also increases with the frequency of the intrinsic oscillation.</p

Finding shortcut and adaptive planning.

<p>(a, b) Turning up the noise level in the external input (indicated below each column) may be a way to prevent the detection of a shortest path. Other parameters as indicated in the caption of <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0127269#pone.0127269.g003" target="_blank">Fig 3</a>. (c) Adaptive planning for a moving goal. A goal at initial goal position (17,7) is moved along the red line after planning time is over and the agent heads from the start towards this goal position. The goal moves one step at each readout cycle. The moving target can reliably be traced (blue line). The planning time is 1 s and the readout time is 250 ms. </p><p></p><p></p><p></p><p><mi>μ</mi></p><p><mi>e</mi><mi>x</mi><mi>t</mi></p><p><mi>g</mi><mi>o</mi><mi>a</mi><mi>l</mi></p><p></p><mo>=</mo><mn>12</mn><mo>.</mo><mn>5</mn><p></p><p></p><p></p>, <i>μ</i>_<i>ext</i> = 12.<p></p