^{1}

^{2}

^{2}

The authors have declared that no competing interests exist.

Using a model for the dynamics of the full somatic nervous system of the nematode

Neurodegenerative diseases such as Alzheimer’s disease, Creutzfeldt-Jakob’s disease, HIV dementia, Multiple Sclerosis and Parkinson’s disease are leading causes of cognitive impairment and death worldwide. Similarly, traumatic brain injury, the signature injury of the Iraq and Afghanistan wars, affects an estimated 57 million people. All of these conditions are characterized by the presence of focal axonal swellings (FAS) throughout the brain. On a network level, however, the effects of FAS remain unexplored. With the emergence of models which simulate an organism’s full neuronal network, we are poised to address how neuronal network performance is degraded by FAS-related damage. Using a model for the full-brain dynamics of the nematode

Understanding networked and dynamic systems is of growing importance across the engineering, physical and biological sciences. Such systems are often composed of a diverse set of dynamic elements whose connectivity are prescribed by sparse and/or dense connections that are local and/or long-range in nature. Indeed, for many systems of interest, the diversity in connectivity and dynamics make it extremely challenging to characterize dynamics on a macroscopic network level.

Of great interest in biological settings is the fact that such complex networks often produce robust and low-dimensional functional responses to dynamic inputs. Indeed, the structure of their large connectivity graph can determine how the system operates as a whole [

Unfortunately, all biological networks are susceptible to pathological and/or traumatic events that might compromise their performance. In neuronal settings, this may be induced by neurodegenerative diseases [

A hallmark feature of damaged neuronal networks is the extensive presence of Focal Axonal Swellings (FAS). FAS has been implicated in cognitive deficits arising from TBI and a variety of leading neurological disorders and neurodegenerative diseases. For instance, FAS is extensively observed in Alzheimer’s disease [

The massive size of human neuronal networks and their complex activity patterns make it difficult to directly relate neuronal network damage to specific behavioral deficits.

More precisely, computational models of

We investigate how network distributed FAS as illustrated in

Details of the underlying neurocircuitry were found by a series of ablation studies, where the functional role of a neuron is evaluated by disconnecting it from the network and observing behavioral deficits [

This commonality suggests that observed behaviors do retain fundamental signatures of the underlying network dynamics. We show such a trajectory for (simulated) motorneuron responses to PLM excitation in

The robustness of the dynamical signatures (population codes) associated with behavior are investigated in injured neuronal networks. Our injury statistics and FAS models are drawn from state-of-the-art biophysical experiments and observations of the distribution and size of FAS.

_{i}.

In a simulated injury, we assign to each affected neuron an axonal swelling from the distribution in

Despite their common statistical distribution, randomly drawn injuries induce qualitatively different changes in the

Distortion of each cycle is quantified via the Procrustes Distance (PD), which compares shapes ignoring translation, rotation or uniform scaling. The PD curves terminate when

Recent experimental work which induced mild TBI in

In

Panel (b) of

Of even greater interest is any possible relationship between injury structure and behavioral output which could, given a specific pattern of distorted dynamics, make predictions about the class of neural injury. To this end, we fit a classification tree to predict the endpoint class from the injury.

The dynamic model for the

The parameter ^{c} the membrane leakage conductance and _{cell} the leakage potential of neuron

In all simulations within this paper, we set

We treat gap junctions between neurons _{i}, which represents the conductivity of synapses from neuron

Here _{r} and _{d} correspond to rise and decay time, and _{i}; _{th}) = 1/(1 + exp(−_{i} − _{th}))). This form of sigmoidal activation is taken from [_{i} depends sigmoidally upon _{i}.

We keep all parameter values from [

Parameters (from [ |
Value |
---|---|

Uninjured Mem. Conductance ^{c} |
10pS |

Uninjured Mem. Capacitance _{H} |
1pF |

Leakage Potential _{c} |
−35mV |

Gap Junction Conductivity |
100pS |

Synaptic Conductivity |
100pS |

Reversal Potential _{j} (Excitatory) |
0mV |

Reversal Potential _{j} (Inhibitory) |
−45mV |

Sigmoidal Width |
0.125mV^{−1} |

Synaptic Rise Constant _{r} |
1 s^{−1} |

Synaptic Decay Constant _{d} |
5 s^{−1} |

Each individual synapse and gap junction is assigned an equal conductivity of ^{c} = 10pS and _{j} will have one of two values corresponding to these classes).

The model is valuable because it generates a low-dimensional neural proxy for behavioral responses. Specifically, constant stimulation of the tail-touch mechanosensory pair PLM creates a two-mode oscillatory limit cycle in the forward motion motorneurons [_{PLML}, _{PLMR} = 2 × 10^{4} Arb. Units for the uninjured model. We take time snapshots these membrane voltages

Note that the single-compartment model which we employ ignores the spatial extent of neurons and specific location of each connection. Our simplified injury model therefore must treat injury as a whole-cell effect. Focal Axonal Swellings (FAS) increase the volume of an axon, which in turn, should alter the cell’s capacitance and leakage conductance within our model. If we approximate a neuron by a single cable of length

When an axon swells, its healthy cross-sectional area _{H} will increase to some swollen value _{i} > _{H}. Thus we assume that the healthy values for capacitance ^{c} will also change according to

We define the individual damage _{i} to neuron _{i} ∝ (_{i} − _{H})/_{H}. Values of _{i} are computed from the experimentally derived distributions in

Mild traumatic brain injuries yield small values of _{i} values assigned from the experimental statistical distributions. The governing equation for an injured neuron is now

We can readily interpret the limiting cases: when _{i} = 0, the original governing equation is recovered, and thus _{i} is large, gap junction and synaptic currents have no effect. The neuron’s voltage decays exponentially to its leakage potential, effectively isolating it from the network.

Note that our random assignment of swelling values neglects any spatial structure of the injury. This could be easily modified by using a distribution which depends on the spatial location of the neuron. Furthermore, this is a very simple model for neuronal swelling, in keeping with our simple model for neurons. It necessarily neglects the actual geometry of swelling. The use of a multi-compartment model would enable this in future studies. Ultimately, there is currently limited biophysical evidence for making more sophisticated models. As such, we have tried to capitalize on as many biophysical observations as possible so as to make a model that is consistent with many of the key experimental observations.

We use MATLAB (version R2013a) to solve the system of neuronal dynamical equations via Euler’s method, using a timestep of 10^{−4}

We classify the resulting injured trajectories as a Fixed Point or a Periodic Orbit according to the following criteria:

Note that these criteria classify very small periodic orbits as fixed points, since their behaviors are very similar. The method may also classify sufficiently slow, long-timescale oscillatory transients as periodic. These tests ignore the first 5 seconds of simulation time (50,000 timesteps), chosen heuristically as a typical timescale of transient decay. After this initial wait, we check the criteria at the end of each subsequent 5 seconds of simulation time until convergence is detected. The results were not observed to be sensitive to the length of this interval.

Stephens et al. [

To interpret the distorted neural activity caused by our simulated injuries, we construct a map from the neuronal activity plane onto the eigenworm plane. The body-shape modes were extracted from Figure 2(c) of [

The behavioral limit cycle in [

It is unclear that the mapping would hold for injured worms, especially without accounting for body-shape modes (eigenworms) from impaired crawling behavior.

We consider only the first two (healthy) behavioral modes. Thus, lack of motion within this plane does not necessarily imply that the worm is not moving. The injured body-shape dynamics could evolve along different modes leaving no traces on the original two.

The lack of direct neuronal analogs for injured network modes limits our ability to interpret arbitrary impaired behavioral responses. Further computational work could also find neuronal proxies for additional behavioral modes so as to enable a more complete mapping. Recent work on blast injuries of worms [

_{A} and _{B} is given by

In other words, it finds the optimal (2D) rotation matrix _{A}. The second shape _{B} is the limit cycle calculated for each injury at the indicated value of

We pre-process the trajectories to extract data points only within a single period. Since injuries usually distort the trajectory length, we use MATLAB’s

We hypothesize that both the injury itself and the PD curves contain meaningful signatures of behavioral outcomes of a given injury. For example, there is always a critical injury level

For these purposes, we classified the endpoints simply by dividing the endpoint distribution along its major axis. Specifically, we take the distribution of endpoints in

We calculate the average PD curve within each class. Since the PD curves may have a different number of points, we first pre-process them. Specifically, we normalize the maximum

We use the

The classification tree that uses normalized PD Curve Shapes to predict the endpoint class yield a cross-validation error of 22.0%. We can compare this to the random case (i.e. the case where PD Curve Shape has no relationship to the class) by repeating this analysis with randomly shuffled class labels. For 100 trials with randomly-shuffled labels, the observed cross-validation error was 43.8 ± 1.4%. Injury vectors were also used to fit classification trees for predicting endpoint classes (see

Thus we can predict (with cross-validated accuracy exceeding 85%) the region into which the endpoint will fall given a specific injury. Moreover, the classification tree in

This study introduces a tractable framework for analyzing how biophysically-inspired injuries distributed across a physical neuronal network induce behavioral deficits. The specific injuries we consider arise from FAS which has been implicated in most leading neurodegenerative diseases, aging and TBI. By identifying low-dimensional population codes within our model which correspond to a known behavior, a proxy metric for cognitive deficit can be constructed. Specifically, limit cycles in our dominant features serve as a neural proxy for actions such as forward motion in the

The ability to provide a theoretical understanding of functional, cognitive and behavioral deficits due to connectomic injuries is a the forefront of TBI and neurodegenerative disease studies. Both have an enormous societal impact and implications. Specifically, TBI is annually responsible for millions of hospitalizations [

Simulated injuries distort dynamical signatures in the network’s activity, such as limit cycles. Our Procrustes Distance metric quantifies how much the shape of the limit cycle is distorted, compared to the healthy cycle. Our results indicate that as different injuries evolve, this metric follows qualitatively different trends (as in

The metrics and methods described in this work can potentially be used to construct diagnostic tools capable of identifying a variety of cognitive deficits. Moreover, the severity of a TBI injury and/or neurodegenerative disease can be quantified by measuring its metric distance from the normal/healthy performance. Our work gives clear mathematical tools capable of formulating such diagnostic tools for assessing injuries and functional deficits.

The present study has many limitations, many due to the lack of biophysical evidence required to build better models. For example, though we treat all neurons as identical passive, linear units, it is known experimentally that different neurons appear to exhibit different behaviors (for example, some neurons appear to be functionally bistable [

We believe the merit of this study lies not so much on the specific results presented, but on the new directions and methodologies it opens for future work. In fact, computational and experimental studies on the effects of network injury are still at their infancy for

Experimental studies would not only test our model, but also in, in conjunction with our work, provide a new testbed for models of injured connectomic dynamics. Our Procrustes Distance metric, shown here to carry information about the eventual outcome of an injury, may also be useful in the real-time analysis of injury progression. Thus our study provides a way forward in monitoring behavioral outcomes of injured networks.

Ultimately at present, limitations in biophysical measurements and neural recordings make it extremely difficult to identify more sophisticated underlying mechanisms responsible for dysfunctions in neural networks, especially when circuits display intrinsically complex behavior and functional activity. We believe the rapid advancement of recording technologies in neuroscience will significantly help refine the model presented here.

Given that the modeling of neuronal networks is one of the most vibrant fields of computational neuroscience [

A

(ZIP)

Figures similar to the rows of

(ZIP)

Videos of injured trajectories mapped onto body shape modes. We include three examples of distorted trajectories along with the healthy trajectory. Similar videos can be created with the included source code.

(ZIP)

MATLAB data file containing injury distributions for all 1,447 trials. Can be used in conjunction with the above source code to recreate the injuries simulated within this manuscript.

(MAT)