Oculomotor control

In human eyes, visual acuity is highest in the foveal region of the retina, and only when the object of interest is held steady on this region. Consequently, the oculomotor system has evolved to perform eye movements that ensure these conditions are met. These eye movements can be voluntary or involuntary, and their role is to fixate the fovea to visual stimuli, to track moving stimuli and to compensate for body movements [1]. Depending on the visual stimuli and the viewing conditions, up to five of the following subsystems are involved in the execution of an eye movement: saccadic, smooth pursuit, vestibular, optokinetic and vergence. The subsystem of interest in this study is the saccadic system. This performs the rapid movements of the eyes (saccades) that transfer gaze from one target to another [2].

Saccades exhibit highly stereotyped behaviour which are commonly summarised using three key relationships between saccadic metrics. The first two relationships state that the saccade duration increases nearly linearly with amplitude and that peak saccadic velocity increases nonlinearly with amplitude at a decreasing rate. These relationships are called the main sequence and have been found to hold for a wide range of normal activities, such as reading, free scene viewing, visual search and walking [3]. The third relationship describes how the velocity profile shape changes with amplitude [4]. It has been shown that the length of the accelerating segment of the velocity profile is roughly consistent for all saccade amplitudes, whereas the length of the decelerating segment increases with amplitude. This causes the velocity profile of small saccades to be symmetrical, whereas those of larger saccades are skewed [4].

Infantile nystagmus

Infantile nystagmus (IN) is an eye movement disorder characterised by involuntary, bilateral, conjugate oscillations of the eyes. The oscillations begin at birth, or shortly afterwards, and are usually restricted to the horizontal plane, although vertical and torsional movements (and combinations of these) have also been observed [5, 6]. Infantile nystagmus can be idiopathic, or associated with sensory defects (e.g. cataract and oculocutaneous albinism) [6, 7]. The estimated prevalence of all forms of nystagmus is 24 per 10000 individuals, of which IN accounts for 14 per 10000 [8]. Nystagmus has a significant impact on the quality of life, with many individuals registered as partially sighted or blind [9]. This makes the development of potential treatments for nystagmus patients an important goal of clinical motor disorder research.

The key characteristics of IN include: the oscillation plane, the waveform amplitude and period, the slow and fast phase, the foveation window and the baseline oscillation (see Additional file 1: Figure S1) [10, 11]. Typically, a slow eye movement (the slow phase) takes the eye away from the target, and a fast eye movement (the fast phase) returns it to the target. An important characteristic of the nystagmus oscillation is the foveation window: this is the cycle interval within which the velocity of the eyes is low enough (< 4 deg/s) for the patient to be able to see clearly [12]. The baseline oscillation is a low frequency sinusoidal oscillation, the amplitude of which is correlated with the nystagmus amplitude [10].

There have been at least 12 different IN waveforms identified in clinical studies [12] and the particular waveform type observed is influenced by factors including gaze angle, stress, attention and arousal [5, 6]. These waveforms can be divided into three groups: pendular, jerk and dual jerk, depending on the waveform shape (see Additional file 1: Figure S2 for schematics of these different groups). Although these waveforms may appear unrelated, it has been shown that they can be described as combinations of different template waveforms, such as sawtooth and pendular oscillations [13]. In pendular nystagmus, slow eye movements move the eye towards and away from the visual target. Jerk waveforms are divided into two categories: unidirectional and bidirectional. In unidirectional jerk, there is one fast eye movement in each period of the oscillation and this is always towards the target. In bidirectional jerk, there are two fast eye movements per period in alternating directions, both of which move the eye towards the target. Unidirectional jerk waveforms are further subdivided into two subcategories: jerk with saccadic foveation and jerk with slow eye movement (SEM) foveation. Jerk with saccadic foveation is subdivided into pure jerk and jerk with extended foveation. Jerk with SEM foveation is subdivided into pseudo-cycloid and pseudo-jerk (see Additional file 1: Figure S2).

Computational models of the oculomotor system

Computational models have proved important in generating and testing hypotheses regarding the mechanisms underlying normal and pathological functioning of the oculomotor system [1, 14–19]. Models based on control theory [14–16] were highly successful in systematically examining the neural substrate underlying the different oculomotor subsystems [1, 17]. A key and influential achievement of this approach was the concept of a neural circuit to convert eye velocity-coded signals into position-coded signals (a neural integrator), derived theoretically from the necessity to hold gaze steady during head movements [20]. The existence of such a neural structure was subsequently verified experimentally [21]. A complementary approach based on nonlinear dynamics emerged towards the end of the 1990s, aiming to understand how the interactions of specific populations of oculomotor neurons could generate both normal and abnormal eye movements. An example of this approach was the model of the saccadic system proposed by Broomhead et al. [22], which was shown to be able to simulate both normal saccades and IN waveforms. A key implication of this model was that IN can be treated as a dynamic disease, in which there is no structural damage to the system and pathological behaviour is instead due to the system’s parameters operating outside of their normal range [23].

Optimising the parameters of computational biology models

Parameter optimisation (or model fitting) is the process of finding the particular combinations of a model’s parameter values that enable it to reproduce the experimentally measured behaviour of the system of interest in an automated fashion. It is a critical step in the construction and analysis of biological models [24], because determining the optimal parameter values enables alternative models to be systematically ranked and experimentally testable predictions to be formulated [24–30]. This allows the assumptions made in constructing a given model to be assessed, and also provides insights into possible modifications to the model to improve the accuracy of its predictions [24, 27–29].

The development of robust, computationally efficient parameter optimisation methods is currently a very active topic of research [31–35]. Indeed, optimisation methods have been used previously to fit the parameters of oculomotor models to experimental data. Cullen et al. [25] were one of the first to employ such methods, using them to build a mathematical model of the neurons that generate saccades. Their methods allowed them to investigate: (i) the dynamic latency of a saccadic neuron; (ii) the prediction capacity of a given model; (iii) whether a more complex model increased the prediction capacity of the model; and (iv) the relationship between the initial conditions and saccade trajectories. Pasquariello et al. [10] used parameter optimisation to fit a model to experimental recordings of the infantile nystagmus baseline oscillation. Their results showed that, on average, the amplitude of the baseline oscillation was half that of the corresponding nystagmus oscillation.

The genetic algorithm

A widely used optimisation method for computational biology problems is the genetic algorithm (GA) [30]. The GA is a stochastic search method based on mimicking natural selection, in which an evolving population of candidate problem solutions (i.e. parameter sets) is used to find the optimal solution (the parameter set yielding the best fit to data) [36]. A fitness function (also referred to as an objective function) is applied to each solution, yielding a value that represents how successful it is in solving the problem (the goodness-of-fit). Based on the values of the fitness function, the GA applies a number of stochastic genetic operators to the population so as to guide the search towards the optimal solution. A major advantage of GAs over deterministic optimisation methods (e.g. gradient descent) is that they do not require any prior assumptions about the objective function to hold, such as continuity or smoothness, with respect to the model’s parameters.

These advantages notwithstanding, when the optimisation problem involves large sets of highly parametrised nonlinear equations being fitted to multiple datasets, evaluating the fitness function for each parameter combination in the population can be very computationally intensive. This effect is amplified by two further factors. Firstly, the stochastic nature of the GA means that multiple runs are necessary to check that the solutions have stably converged [36]. Secondly, the GA itself has a number of parameters which require tuning, including those specifying the fitness function. It is therefore necessary to systematically test different combinations of GA parameters to ensure that true optima are returned. Consequently, considerable efforts have been made to reduce the execution time of GAs. For a given set of GA parameters, the same fitness function and genetic operators are applied to the solutions in the population. This makes GAs very suitable to parallelisation and many parallel architectures have been developed to exploit this feature to achieve acceleration [36].

However, for problems involving realistic parameter numbers, high-performance computing (HPC) clusters (i.e. multiple connected computers) are required to obtain results within a reasonable timeframe. Although HPC clusters provide significant processing power, and the intrinsic parallel architecture of GAs allows them to be executed over the multiple central processing units (CPUs) within a cluster, they also have a number of disadvantages. Chief amongst these are their purchase price, as well as the high costs associated with maintenance and energy consumption. An alternative approach is to use consumer-level graphics processing units (GPUs). These possess parallel computing power that can be comparable to that of the CPUs in a small HPC cluster. Furthermore, GPUs have a considerably lower cost and require substantially less maintenance. Recently, parallel GAs have been developed that run on GPUs, yielding significant speedups compared with CPUs [37–39].

Multi-objective optimisation

Here, x=(x ₁,…,x _n ) is a vector which gathers together the parameters (decision variables) of the model and \(\left \{f_{i} : \mathbb {R}\rightarrow \mathbb {R}, \, 1\le i \le m\right \}\) are the objective functions which quantify the quality of the model fit to different experimental features for each choice of x. Finally, the functions \(\left \{g_{j}:\mathbb {R}\rightarrow \mathbb {R},\, 1\le j \le l\right \}\) impose any relevant constraints on the problem (e.g. that the parameters are within a biologically feasible range) and {b ₁,…,b _l } are the corresponding constraint values [41–43]. Optimal solutions of (1) thus correspond to parameter sets that give the best fits to the experimental data, within the bounds specified.

One approach to solving such problems is the weighted-sum method [44], in which the objective functions f _i (x) in (1) are summed with appropriate weights w _i to give a scalar objective function \(\sum _{i} w_{i} f_{i}(\mathbf {x})\) to be minimised (uni-objective optimisation). However, this requires the w _i values that maximally exploit the information provided by the different objectives to be specified and these are typically not known a priori [41, 42, 45]. An alternative approach is the ε-constraint method, in which the MOO problem is reduced to a uni-objective problem by minimising one of the objectives and setting the others to be constrained to specific values [40, 46]. Different constraint values for each objective have to be explored in order to obtain solutions that provide a desirable trade-off between the objectives.

Other MOO methods are based on the concept of Pareto dominance, where a solution x is said to dominate a solution x ^′ if and only if the following two conditions hold: first, solution x is no worse than x ^′ on all objectives (f _i (x) ≤ f _i (x ^′) for all i) and second, solution x is better on at least one objective (f _j (x) < f _j (x ^′) for some j) [41, 42, 45, 47, 48]. The algorithms return an estimate of the set of solutions for which none of the objectives can be improved without compromising one of the others (the set of non-dominated solutions). These Pareto optimal solutions (the Pareto set) represent the optimal trade-off between the individual objectives. The image of this set in objective space is referred to as the Pareto front. An effective multi-objective optimisation algorithm returns an estimate of the Pareto set for which the corresponding front is close to the true one and on which the individual points are well separated in objective space [41, 42, 45, 48]. A good spread of solutions facilitates the selection of a final solution from the Pareto set. The method used for this final selection procedure is generally dependent on the particular problem of interest and often utilises the values of the Pareto optimal solutions on all objectives [49, 50].

MOO methods have been used previously in a range of bioinformatics and computational biology applications, such as RNA inverse folding [51], analysis and design of gene regulatory networks [52–54], and neural model fitting [55] (see [56] for a comprehensive review). A key advantage of MOO methods is that the structure of the Pareto set provides information about the conflicts between the objectives and the distribution of the possible solutions, which can yield useful insights into the particular biological system being modelled [56]. Among the most widely used MOO methods are multi-objective genetic algorithms (MOGAs). A very popular MOGA is the Non-dominated Sorting Genetic Algorithm II (NSGA-II) [45, 57]. The main characteristics of NSGA-II are elitism, a fast non-dominated sorting approach and a selection operator that ensures better spread of the solutions and better convergence near the Pareto front. NSGA-II has been successfully applied to a broad range of optimisation problems in different application areas [58–60].

Aims and overview

The first aim of this computational study was to optimise the saccadic model of Broomhead et al. to existing experimental recordings of two types of eye movements: (i) infantile nystagmus oscillations; and (ii) horizontal saccades recorded from healthy subjects. Both optimisation problems are multi-objective, as conflicting objectives must be optimised together to reproduce key experimental characteristics. In the case of saccades, we optimised the model to experimental data measured for three different saccade amplitudes, treating each dataset as a separate objective. For nystagmus, we used the shape and period of each waveform as our objectives. We developed novel multi-objective fitness functions for each optimisation problem, and used NSGA-II to find Pareto optimal solutions in each case. Synthetic datasets generated from the model were used to help tune the NSGA-II parameters. The optimal solutions obtained in this fashion enabled us to quantify which experimental behaviours could be replicated by the model, thereby identifying possible modifications to increase the model’s predictive capacity.

The second aim of the study was to accelerate NSGA-II by parallelising its most computationally intensive component – the numerical integration of the oculomotor model – and executing the algorithm on a CPU-GPU combination. We show that for the optimisation of the oculomotor model (and any similar computational biology model), this approach can yield an order of magnitude decrease in computation time. Furthermore, we conclude that the selection of other NSGA-II processes for parallelisation in order to further improve performance will be dependent on the population size used.

The oculomotor model

The model was constructed on the basis of neurophysiological studies showing that the neural signal controlling saccades is generated by excitatory burst neurons (EBNs) in the brainstem [61]. This velocity-coded control signal (the pulse) is converted into a position-coded signal (the step) by the neural integrator. The pulse and step are then relayed to the extraocular muscles (the muscle plant), which move the eye to the desired position. Eqs. (2) and (3) model this muscle plant as a second order, overdamped linear system with time constants T ₁=0.15s and T ₂=0.012s [15, 62]; g and v represent the horizontal eye position (gaze angle) and eye velocity, respectively. Eq. (4) models the neural integrator signal n; this is characterised by the long time constant T _N =25s, which is necessary to hold the eye steady at the target position following the saccade [62].

which was derived from direct fits to experimental data (single-cell recordings from alert primates) [61].

The parameters α ^′ and β ^′ in (8) quantify the neurons’ response to error signals generated by motion in the direction of maximal firing (the on-response). α and β quantify the response to motion in the opposite direction (the off-response); the off-response causes a slowing of the eye towards the end of the saccade to prevent overshoot of the target and is referred to as the braking signal [61]. The parameter ε determines the speed at which the EBNs respond to the error signal, with smaller ε values yielding faster responses. Finally, (7) is the equation for the motor error. This was derived by assuming that m=Δ g−s, where Δ g=T−g is the eye displacement necessary to attain the target position T, and s is an estimate of eye displacement generated by a resettable integrator – separate to the NI – which also integrates the saccadic velocity command r−l [22].

In [22], Broomhead et al. empirically selected the values of the parameters α ^′, β ^′ and γ to match the saccadic main sequence and then examined the effect of varying α (off-response magnitude), β (off-response range) and ε (burst neuron response time). This manual parameter search revealed that the model is capable of generating accurate and inaccurate saccades, as well as several infantile nystagmus oscillations (see Additional file 1: Figure S3).

Data

We used two types of data for the model optimisation: experimental and synthetic. The experimental data was taken from previous studies, and comprised a collection of nystagmus oscillations recorded during viewing at rest, together with mean saccadic velocity profiles recorded for different saccade amplitudes. The synthetic data was generated de novo from the saccadic model itself. Prior to optimising the model to experimental data, we performed fits to synthetic timeseries to assess: (i) how effective the multi-objective fitness functions were in reproducing experimental eye movement characteristics; and (ii) how the convergence and accuracy of the optimisation procedure varied with the NSGA-II parameters, thereby facilitating the final selection of these parameters. In particular, we checked that NSGA-II was consistently and reliably converging to the model parameters used to generate each synthetic dataset.

Nystagmus oscillations. The experimental nystagmus database from which the data used in this study was taken consisted of horizontal eye movement recordings from 48 nystagmus patients enrolled on a drug trial [63]. From this database, data from 20 idiopathic subjects was chosen. The remaining 28 subjects in the drug trial had additional medical conditions and hence their data was rejected. The idiopath group consisted of 13 males and 7 females, with a mean age of 40.05±8.29. The eye movement recordings we used were those obtained before any drugs were administered to the participants. Binocular eye movements were recorded with the head stabilised by a chin rest. The sampling rate of the recordings was 250 Hz, with a resolution of 0.005 degrees. Typically, subjects had irregular waveforms with 4-6 cycles per second. An example of one of the nystagmus time series from the database is shown in Fig. 1.

The model parameters used to produce the synthetic IN waveforms are given in Additional file 1: Table S1; the waveforms themselves are shown in Additional file 1: Figure S4. These particular oscillations were chosen as fitting targets because they represent a broad distribution of amplitudes and periods.

Saccadic velocity profiles. Experimental saccade data was taken from the study of Collewijn et al. [64]. The model was fitted to the mean velocity profiles reported in [64] for horizontal saccades of 5, 10 and 20 degrees (see Fig. 2). These particular saccade amplitudes were chosen because they are in the range of most frequently generated saccades [65, 66].

Synthetic saccadic velocity profiles were generated by the model using the parameter values listed in Additional file 1: Table S2; the velocity profiles themselves are shown in Additional file 1: Figure S5.

The optimisation protocol

The goal of the optimisation protocol we developed was to find values of the six parameters { α,β,ε,γ,α ^′,β ^′} of the oculomotor model (2)-(7) that enabled it to reproduce the experimental eye movement recordings of interest. In the case of saccades, the model was fitted directly to experimental saccadic velocity profiles for each saccade amplitude. For nystagmus data, the model fitting involved finding simulated waveforms with the same amplitude, period and shape as the experimental data. For both optimisation problems, we employed the MOGA NSGA-II [57], using the implementation of the algorithm included in the MATLAB Global Optimisation Toolbox [67]. A key reason for our choice of optimiser was that NSGA-II has been successfully applied to optimisation problems with complex objective spaces [58–60]. Akman et al. [18] showed that the oculomotor model (2)-(7) has a rich bifurcation structure, incorporating Hopf, homoclinic, saddlenode, pitchfork and gluing bifurcations [68]. These bifurcations can create sharp changes in waveform type under small parameter variations, significantly compromising the ability of a standard optimiser to efficiently explore the parameter space.

Formulating the fitness functions and selecting the NSGA-II parameters

In order to fit the model to the synthetic and experimental datasets, a fitness function for each data type (nystagmus oscillations and saccades) had to be formulated that measured the goodness-of-fit. When using a GA as an optimiser, different candidate fitness functions should be tested in order to assess the extent to which they are able to successfully quantify the target data characteristics [69]. The formulation of the fitness functions was a trial-and-error method. Each time we created a new fitness function (or modified an existing one), we tested it on the synthetic and experimental datasets to assess the effect of the change on the fitness function landscape and the quality of solutions obtained. For the nystagmus data, we initially used the squared difference between simulated and target waveforms. However, we found that a bi-objective fitness function based on oscillation period and shape gave better results. This observation is consistent with studies showing that multi-objective optimisers are less likely than uni-objective optimisers to become trapped in local minima [70, 71]. For the saccade data, we initially chose a multi-objective fitness function based on the saccadic main sequence. However, we found that this gave poor fits to the experimental data. We therefore formulated a multi-objective fitness function based on the squared difference between the experimental and simulated velocity profiles for different saccade amplitudes. The final versions of the fitness functions we used are described in greater detail in the “Fitness function for nystagmus waveforms” and “Fitness function for saccadic velocity profiles” sections below.

As mentioned previously, NSGA-II has parameters of its own (e.g. population size, mutation rate/type, crossover type, selection type) that affect the performance of the optimiser, as assessed by key benchmark measures such as the fitness value of the best solution and execution time. However, there are no general rules governing which choice of NSGA-II parameters is best for a given problem and so different combinations of these should also be tested [72]. Typically, given the stochastic nature of the GA, the goal is to find combinations of the GA parameters that allow it to converge stably to minima of the optimisation problem over multiple GA runs [36].

To quantify the accuracy and convergence properties of the optimal solutions found by NSGA-II, we used a hypervolume indicator. This was defined in terms of the volume \(\mathcal {H\left (\hat {\mathcal {F}},\mathbf {y}_{R}\right)}\) in objective space formed between the estimated Pareto front \(\hat {\mathcal {F}}\) and a fixed reference point y _R that was chosen to be dominated by all the points in \(\hat {\mathcal {F}}\) [73]. To calculate an accurate approximation to this area we used the MATLAB function convhull.m. We compared this quantity to the volume \(\mathcal {H}\left (\mathbf {0},\mathbf {y}_{R}\right)\) of the hypercuboid formed between y _R , the origin 0 of the objective space and the points obtained by projecting y _R onto each objective axis separately. The ratio \(\mathcal {H}\left (\hat {\mathcal {F}},\mathbf {y}_{R}\right)/\mathcal {H}\left (\mathbf {0},\mathbf {y}_{R}\right)\) of these two quantities provided a means for assessing whether the Pareto front estimates for a specific choice of fitness function and NSGA-II parameters were converging with generation number. Due to the stochastic nature of the optimiser, for each such choice, we calculated the mean and variance of this ratio over 16 independent optimisation runs, fixing the reference point y _R to be one dominated by all non-dominated sets produced collectively over the individual runs. In addition, to provide a measure that was more convenient for the visualisation of results, we subtracted the ratio from 1. The resulting measure \(\mathcal {H}_{I}=1-\mathcal {H}\left (\hat {\mathcal {F}},\mathbf {y}_{R}\right)/\mathcal {H}\left (\mathbf {0},\mathbf {y}_{R}\right)\) was taken as the overall goodness-of-fit, with a value of 0 indicating a Pareto front comprised solely of the origin: i.e. a solution that perfectly fits all objectives simultaneously. The accuracy and convergence of the results were further evaluated by assessing how the minimum Euclidean distance \(d_{\hat {\mathcal {F}}}=\min \left \{||\mathbf {y}||_{2}:\mathbf {y}\in \hat {\mathcal {F}}\right \}\) between the estimated Pareto front and the origin of objective space varied with population size (a minimum distance close to 0 indicates a good fit).

simulating a saccade of amplitude Δ g initiated from rest in the primary position (0 degrees – looking directly ahead). For nystagmus fitting, Δ g was set to 1.5, simulating a rightward saccade of 1.5 degrees, while for saccade fitting, Δ g was set to each of the different saccade amplitudes (5, 10 and 20 degrees) in turn.

The NSGA-II parameters that we varied to explore optimisation performance were as follows: the population size (values tested were 500, 1000, 2000, 4000 and 8000); the crossover function (intermediate or heuristic) and the distance measure type (phenotype or genotype). We did not vary all the NSGA-II parameters as this would have increased the computation time considerably, making the process impractical. Accordingly, in order to keep the running time short, the number of generations was fixed at 100 for saccade and synthetic nystagmus fits and at 200 for experimental nystagmus fits. The selection type was set to binary tournament (this selects each parent by randomly drawing two individuals from the population and then chooses the one with the highest fitness for crossover), and the mutation type was set to adaptive feasible (this mutates individuals towards regions in parameter space which – based on previous generations – are more likely to give better fitness).

It was expected that the population size would be a key determinant of NSGA-II accuracy and convergence, since if the population size is too small, the algorithm is unable to converge to the Pareto front, whilst increasing the population size requires greater computational resources [74]. Hence, when choosing the population size, a balance needs to be found between accuracy/convergence and the resulting computational load (the significant increase in computation time with population size can be seen for synthetic nystagmus data in Additional file 1: Table S3). Final population sizes were chosen to provide a reasonable trade-off between accuracy/convergence and running time. In the case of the crossover method, we found that heuristic crossover returned good results for considerably smaller population sizes compared with intermediate crossover. Of the remaining NSGA-II parameters, best results were obtained with the phenotype-based distance measure.

Fitness function for nystagmus waveforms

For nystagmus waveforms, the multi-objective fitness function evaluated each individual to extract two objectives: (i) the difference in shape between the target and the scaled-to-target-period simulated waveforms; and (ii) the difference in period between the waveforms. An amplitude comparison was not included, as exploratory NSGA-II runs on both synthetic and experimental waveforms indicated that it was redundant: the shape comparison was sufficient for the optimiser to yield solutions with the correct amplitude. The input to the fitness function was a single period of the target waveform and a single period of the simulated time series.

Extracting a single period of a simulated waveform. The procedure for extracting a single period of a simulated oscillation – either for use as a target waveform or for fitting – was as follows. First, the model was integrated for 6 s. This time interval was chosen because exploratory runs of the method showed that it enabled one period of the waveform to be reliably extracted across a broad range of parameter values. The initial 2.4 s of the time series was then discarded as transient dynamics preceding the onset of oscillations. Next, the remaining portion of the time series was normalised to lie in the interval [0,1] and the local minima of the oscillation were identified. A single period of the waveform was then obtained by extracting all the points with value below 0.2 and between the last two local minima in the time series. If no such points were found, then the input was classified as non-oscillatory and the process terminated, with the individual assigned an arbitrarily high value (10⁶⁰) for each objective.

Extracting a single period of an experimental nystagmus recording. For experimentally recorded waveforms, the extraction of a single period was much more challenging due to the nondeterministic nature of real nystagmus oscillations, in which each successive cycle has a slightly different period, amplitude and shape (see Fig. 1). This raises the question of which cycle(s) should be chosen for the fitting procedure. To address this, we used a variant of the nonlinear time series analysis method developed by So et al. [75, 76]. Starting from a scalar sequence of system measurements, this allows one to find either unstable periodic orbits (UPOs) generated by a deterministic system, or stable periodic orbits which have been destabilised by a noise process [75–79]. The method is based on the assumption that there is a significant deterministic component to the experimental system of interest, and so the extracted periodic orbits can be used to characterise the system’s dynamics by providing single oscillation cycles that are representative of the observed nonperiodic waveforms [23, 77]. This method has been previously applied successfully to nystagmus recordings [78, 79]. Here, we used the version of the method developed by Theodorou et al. [79], which we describe below.

The first stage of the method involves identifying the intervals { τ ₁,τ ₂,…,τ _N } between individual nystagmus cycles by thresholding the velocity time series. The velocity threshold was chosen to correspond roughly to the middle of the fast phase of each nystagmus waveform. Next, we applied the method of delays [80, 81] to the interval data. In the method of delays, a sliding window of d samples is moved through the data, generating a sequence of d-dimensional vectors. Given the interval time series {τ ₁,τ ₂,…,τ _N }, the delay vectors {w ₁,w ₂,…,w _N−d+1} are defined as w _k =(τ _k ,τ _k+1,…,τ _k+d−1) ^T . The evolution of the delay vectors is governed by a nonlinear map F, via w _k+1=F(w _k ), which – under certain genericity conditions – is related to the corresponding map governing the evolution of inter-cycle intervals in the underlying experimental system by a smooth, invertible change of coordinates [80, 81]. This means that fixed points of F, which are points in the delay space w _∗ for which F(w _∗)=w _∗, correspond to periodic cycles in the ambient system. For such points, τ ₁=τ ₂=⋯=τ _N =τ _∗; fixed points therefore lie on the long diagonal of the delay space (the set of d-dimensional vectors with equal entries). Candidate fixed points can therefore be identified by looking for delay vectors that lie close to this diagonal.

In Eq. (11), I is the d×d identity matrix and D F(w _n ) is the d×d Jacobian matrix of F evaluated at w _n , which is estimated from the delay vectors using finite differences [79]. The transformed data lying within a small cross-sectional tube along the delay space diagonal is then projected onto the diagonal itself. Fixed points are identified as sharp peaks in the histogram obtained by binning the 1-dimensional projected data [76].

To extract UPOs from the nystagmus recordings using this method, we chose the embedding dimension d to be 2, the diameter of the cross-sectional tube to be 0.5 units and the histogram bin size to be 0.025 s [78]. Candidate period orbits were then identified as those with inter-cycle intervals τ _k lying within 0.0125 s of the histogram peak τ _∗. From this collection of periodic waveforms, we choose the cycle with the smallest difference between its first and last points. Because it was more convenient to use waveform targets that start at the beginning of the fast phase, we concatenated three copies of the waveform and chose the section between two successive fast phases as the final cycle to be fitted. An example of the application of this procedure to an experimental nystagmus waveform is shown in Additional file 1: Figure S6.

where S _i and T _i denote the values of the scaled and target waveforms, respectively, at time t _i , where t _i+1−t _i is the (fixed) data sampling interval.

where τ _E is the period of the extracted, unscaled waveform and τ _T is the period of the target waveform.

Fitness function for saccadic velocity profiles

Here, S _ij and T _ij denote the values of the simulated and target profiles, respectively, at time t _i , where t _i+1−t _i is the (fixed) data sampling interval and j indexes the amplitude. To extract the simulated velocity profile that was compared to the target one, we set the simulated velocity profile’s end point to that of the target profile, thereby ensuring that both had the same length.

Selection of the best individual

When NSGA-II terminates, it returns a set of points that approximates the Pareto front. For nystagmus fitting, to obtain a single best-fit parameter set from this front, we selected the individual with the smallest difference in period to the target waveform, as this consistently provided a very good shape fit as well. We also explored the effect of selecting the individual yielding the lowest value on each objective separately (i.e. we chose the solutions from the Pareto set giving the best fits to saccades of 5, 10 and 20 degrees).

Acceleration of the optimisation method using GPUs

In order to reduce computation time, we utilised the parallel capabilities of GPUs using a parallel implementation based on the master-slave model [36, 82]. In this implementation, for each instance of NSGA-II, the most computationally intensive task is distributed to multiple processors (the computing units of the GPU), while a master processor (the CPU) is used to control the parallelisation and run the remaining tasks.

In designing our optimisation protocol, we used the MATLAB profiler – which measures the execution time of each MATLAB function – to compare the time required to integrate the model to that required to perform the NSGA-II operations and evaluate the fitness function. Additional file 1: Figure S7 shows the mean execution time of each task, calculated from 8 independent NSGA-II instances run for 5 generations each with different population sizes. It can be seen that for all population sizes tested, the model integration was the most computationally intensive task. Hence, for our application, we chose to implement this process only on the GPU, with the genetic operations and fitness evaluation being performed by the CPU. Moreover, we employed the independent runs parallel model [36], in which multiple master-slave NSGA-IIs are executed independently on the CPU-GPU combination, with no communication between them.

Running multiple NSGA-II instances in parallel. To implement the parallelisation outlined above, we used the program organisation shown schematically in Fig. 3. The general idea behind the method is as follows: an NSGA-II manager program (written in MATLAB), which runs on the CPU, creates the individual NSGA-II MATLAB instances that run on the CPU in parallel. Each of these NSGA-II instances sends the parameter combinations (individuals) comprising its population to the GPU server program (another MATLAB instance), which writes these combinations to an input file. This file is read by the GPU executable (described in Additional file 1), which calculates the oculomotor model solutions. The results of the GPU executable are written to a binary file that is split by the GPU server program into multiple binary files containing the solutions corresponding to each NSGA-II instance. Finally, the NSGA-II instances read the binary files, evaluate the fitness of each individual in the population and then apply genetic operations to create the new population.

Running multiple instances of NSGA-II caused two main problems, due to the very large files that were generated. In particular, writing to (and reading from) the hard disk takes a considerable amount of time compared to writing to the RAM, thereby creating a speed bottleneck. To address this, we used a RAM drive – a block of the system RAM that the software uses as a disk drive – which proved to be considerably faster.

Hardware used. We tested our executable on four GPUs and one CPU. The GPU models were the AMD FirePRO W8100, the AMD Radeon HD 7970 GHz Edition, the NVIDIA Tesla K20m and the NVIDIA Tesla K40c. The CPU model was the Intel Core i7-4790K CPU. The reason we chose these particular GPUs is that they provide high double-precision compute capabilities which are required for the integration of stiff ODE systems. We chose to compare the performance of the above GPUs to the Intel Core i7-4790K, as it is one of the high-end CPU models currently available.