Neural models

Hodgkin [29] distinguishes between types of neuron responses. The first type of neuron (Type I) always responds to small depolarization by advancing the next spike. An example of such a neuron is the integrate-and-fire model. The second type (Type II) is exemplified by the Hodgkin-Huxley model in which there is a negative region just after the refractory period, where a depolarization delays the firing of the next spike because the delayed rectifier potassium current is greater than the sodium current, while an excitatory post-synaptic potential received at a later time advances the firing. In this paper both Type I and Type II models are assessed.

Quadratic integrate-and-fire neurons

The QIF model [30] displays Type I neuron dynamics [31]. The time evolution of the neuron membrane potential is given by:where V is the membrane potential, with V_r and V_t being the resting and threshold values respectively. C is the capacitance of the cell membrane. τ is the membrane time constant such that τ = RC with R being the resistance. I represents a depolarizing input current to the neuron.

An action potential occurs when V reaches a value V_peak at which point it is reset to value V_reset. The QIF model is equivalent to the theta neuron model described by Ermentrout and Kopell [32] if one sets the reset condition V_peak = ∞ and V_reset = −∞. Like Börgers and Kopell [33] we use values V_r = V_reset =0 and V_t = V_peak = 1, which reduces the above equation to:

Here and is set to the value 2 for all experiments carried out in the paper. When working with the QIF model we assume a membrane potential between V_r = −65 mV and V_t = −45 mV.

Hodgkin-Huxley neurons

The Hodgkin-Huxley [34] model is widely considered as the benchmark standard for neural models. It is based upon experiments on the giant axon of the squid. Hodgkin and Huxley found three different types of ion current: sodium (Na⁺), potassium (K⁺), and a leak current that consists mainly of chloride (Cl⁻) ions. Different voltage-dependent ion channels control the flow of ions through the cell membrane. From their experiments, Hodgkin and Huxley formulated the following equation defining the time evolution of the model:

C is the capacitance and n, m and h describe the voltage dependent opening and closing dynamics of the ion channels. The maximum conductances of each channel are: g_k = 120, g_Na = 36 and g_L  = 0.3. The reversal potentials are set so that that E_k = −12, E_Na = 115 and E_L = 10.6. The rate functions for each channel are:

All work in this paper using the HH model adjusts the neuron resting potential from 0 mV of the standard HH implementation to the more accepted value of 65 mV [35].

Synaptic model

The synaptic model for simulations using the QIF model is a current synapse that simply multiplies the incoming spike by a synaptic weight:where I_j(t) is the input to neuron j and time t. w_ij is the synaptic weight from neuron i to neuron j, and d_ij is the synaptic delay from neuron i to neuron j. A list of the all n spikes produced from neuron i during a simulation are denoted by their spike times t_i,k, where k = 1,2…..n. δ is a delta function applied to t-d_ij-t_i,k, such that adjusting the current time t by the synaptic delay d_ij identifies the spike production time at neuron i for which a spike is due to arrive at neuron j at time t. If t_i,k matches this spike time then the delta function produces an output value 1.

The HH model uses conductance synapses, and so uses reversal potentials to further scale incoming spikes. The latter model is as follows:

The additions to the previous synaptic model are, Rev which is the reversal potential, and V_j, which is the voltage of the target neuron. The reversal potentials for the model are set to the same values in all experiments. For excitatory inputs the reversal potential is set to 0 mV, and for inhibitory inputs the reversal potential is −70 mV. Not using a synaptic reversal model for the QIF model is equivalent to using a synaptic reversal model with reversal potentials set to +∞mV for excitatory neurons and -∞mV for inhibitory neurons.

Evolution of the architecture for neural oscillatory nodes

Groups of neurons firing together rhythmically can occur because of intrinsic firing patterns of excitatory principal cells or due to common input from a pacemaker, however, it is more common both in the cortex and the hippocampus that rhythmic firing happens as an emergent property of interactions between excitatory principal cells and inhibitory interneurons. Variations of this mechanism, known as pyramidal inter-neuronal gamma (PING), can give rise to both faster gamma oscillations as well as slower oscillations such as theta in the cortex and the hippocampus [3].

Excitatory neurons drive the entire local network, including inhibitory interneurons. The most strongly driven inhibitory neurons will fire first and provide inhibition to numerous other inhibitory neurons. The inhibitory effect on all these neurons will disappear at approximately the same time. Affected inhibitory neurons will then fire roughly together, causing large numbers of inhibitory neurons to be entrained to a rhythm within just a few oscillatory cycles [36]. This rhythmically synchronized inhibition also affects the network's excitatory neurons with a fast and strong synaptic input [37] thus leaving only a short window for the excitatory neurons to fire after one period of inhibition wears off and before the next one starts [38].

The oscillators used in this work conform to a PING architecture. Whilst the general PING architecture is well understood, the specific details required for both particular oscillatory frequencies and neuron model vary and involve a large space of parameter values within the general PING framework. In order to provide a wide range of different intrinsic oscillatory frequencies for the neural oscillator nodes used in the experiment, it was decided to obtain these parameter values by use of a genetic algorithm (described below). The genetic algorithm evolved, within biologically plausible bounds, every oscillatory frequency between 30 Hz and 50 Hz for both QIF and HH models. The evolutionary mechanisms were constrained so that each neural network was evolved in accordance with the general PING architecture mentioned above. All neural populations for the PING oscillators used an excitatory layer of 200 neurons and an inhibitory layer of 50 neurons. The excitatory layer drives the entire network and so is the only one to receive external input. The input is generated from a Poisson process with parameter λ = 4.375. For QIF models the inputs were scaled by 8 and for the HH models the inputs were scaled by 15 in order to provide sufficient stimulus to induce firing. The networks were wired up with connections between inhibitory neurons (II), from excitatory to inhibitory neurons (EI) and from inhibitory to excitatory neurons (IE). Excitatory to excitatory (EE) connections were excluded in order to limit saturation effects (by which we mean all neurons firing all of the time). Saturation effects tend to arise in the later simulations in which many neural PING nodes were wired together. The possibility of saturation was not otherwise catered for in the evolutionary process due to the PING networks being evolved in isolation. The PING architecture used is illustrated in figure 1a. In addition to the synaptic weight, a scaling factor of 5 was used on all synaptic current in the oscillatory populations for both QIF and HH models to simulate networks of a larger size than we could feasibly simulate, given the number individuals in a population and the number of generations in an evolutionary run, as well as the large number of simulation runs using 10 neural PING nodes in our final experiment.

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Figure 1. PING oscillatory architecture and behaviour.

(A) The pyramidal inter-neuronal gamma (PING) architecture used for the neural oscillator nodes in the simulation experiments. To generate oscillator nodes of different frequencies for different neural models this base architecture was used with a genetic algorithm evolving the weights and delays for the synaptic connections. (B) Example of the firing behaviour of an evolved QIF PING node oscillating at 30 Hz.

https://doi.org/10.1371/journal.pone.0062234.g001

A genetic algorithm is a blind search and optimisation technique based upon the theory of natural selection [39]. Parameters are encoded in a pseudo genome, and are used to instantiate an individual, in this case a neural network. A population of individuals are tested and scored for their fitness at performing the test. Based upon their fitness ranking pairs of individuals are chosen to produce offspring for the next generation via crossover of their genomes. Mutation is then applied to some parameters in the new offspring genome. As this process continues over generations individuals in the population become optimised at performing the target task. The parameters that were evolved in this work were the synaptic weights and delays. Both of these were generated during genome expression of each individual in each generation using an approximately normal distribution, with the means and variances for the weights and the delays being the parameters in the genome evolved. The distribution is approximately normal as the weights were bound to evolve values between 0 and 1 for excitatory connections and 0 and −1 for inhibitory connections. Delays were similarly bound. Long delays are quite unrealistic for a cluster of neurons in which all neurons are anatomically close together. In the cortex synaptic latency ranges from 0.2 ms to 6 ms [40]. In order to produce realistic results, excitatory delays were bounded between 1 ms and 10 ms. The IE and II delays were allowed to have a maximum value of 50 ms to simulate the effect of slow inhibitory interneurons, the behaviour of which was otherwise not modelled.

Individuals were tested for 5000 ms of simulated time in which they received external input to the excitatory layer as described above. The fitness function for the genetic algorithm consisted first of taking the spike firing times of the excitatory population and converting it to a continuous time-varying signal. This was achieved by binning the spikes over time, and then passing a Gaussian smoothing filter over the binned data. Next a Fourier transform was performed on the mean centred signal to produce the frequency spectrum of the signal. The main fitness term was calculated by creating a scaled Gaussian centred around the desired frequency f in the spectrum of the form:

The frequency spectrum s was subtracted from this and normalized:

An extra penalty term was introduced to discourage frequencies outside the desired range. This was achieved by multiplying the frequency spectrum by −0.002 in the areas further away from the desired frequency whilst ignoring the area at and immediately around the desired frequency. The result was then normalized and added to the main fitness term.

The evolutionary population consisted of 20 individual genomes. For each generation, each individual was tested for 5000 ms of simulated time. After this each individual was rated for fitness and probabilistically selected for the next generations' parents based upon their fitness ranking. Crossover was performed on parent genomes after which mutation was applied to the offspring with a probability of 0.1.

All evolved weights for QIF solutions had very high means and small variances, whereas the HH solution showed greater variation in the weight means across evolved solutions for different frequencies, indicating greater sensitivity in the model and solution in that they require a very specific balance of the parameters for each particular solution. The means for the delays evolved for both QIF and HH solutions had a similar form, from which can be concluded that the EI mean delay+IE mean delay≈1000/2f. Figure 1b shows a raster plot of an evolved PING oscillator with regular bursts of firing in the excitatory layer at 30 Hz.

Extraction of intermittent frequency strands

The work presented in this paper aims at assessing the correlation between the fluctuating frequencies in different neural oscillators that are connected together in a network. In order to achieve this we first need to extract the instantaneous frequency responses for each neural oscillator at each moment in time during a simulation. The standard techniques for doing this are to either use a short-time Fourier transform or a wavelet transform. To perform either first requires converting the firings of an oscillatory neural population into a continuous time signal upon which one of these transforms can be performed. We only use the excitatory layer in a neural oscillator when producing this signal. The signal is obtained by first binning the number of spikes at each moment in time for the excitatory layer, and then passing a Gaussian smoothing filter over the data. Finally the signal is centred around its mean to obtain the continuous time signal upon which we can perform the transform.

Both Fourier and wavelet based approaches for extracting the time-frequency information from a signal suffer from shortcomings due to the time-frequency uncertainty principle. A Gabor wavelet transform has been chosen for use in this work, because the responses of Gabor wavelets have optimal properties with respect to the time-frequency uncertainty principle [41]. The Gabor wavelet used had a centre frequency of 0.6 Hz and was applied with a continuous wavelet transform using scales from 1 to 100 in increments of 0.1, and a delta of 0.001. Figure 2a shows the scalogram of a wavelet transform taken from the excitatory layer of a neural oscillator in one of the experiments. In this experiment, as in all others, the oscillator was placed in a network with 9 other oscillators, each oscillating at a different intrinsic frequency. There is a given probability of connecting each oscillator to another in the network, and a given synaptic connection weight for the connections formed between oscillators. Both connection probability and synaptic weight are unique to each experimental run. Figure 2b shows a raster plot of the firing behaviour of the same excitatory layer between time points 1000 ms and 1500 ms in the simulation. It can be seen that the spacing of the bursts of firing between 1050 ms and 1150 ms is wider and thus at a different slower frequency to the spacing of the bursts of firing between 1200 ms and 1275 ms. The wavelet response to the slow and then faster bursting can be seen as a difference from low to high frequency on the scalogram in the same temporal area and around the frequency range from 30 Hz–50 Hz. These responses are deviations from the regular 33 Hz bursting that the PING oscillator was evolved to fire at and are due to the interaction with the other oscillator nodes.

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Figure 2. Extraction of frequencies from population firings (A) Scalogram of the excitatory layer of a neural PING node that has been connected to 9 other nodes each oscillating at a different frequency.

(B) Firing behaviour of the same excitatory layer between 1000 ms to 1500 ms in the simulation. Note how the spacing in between the burst of firing is reflected as different frequencies in the scalogram in panel A. (C) Time slice of the scalogram in panel A taken at 1460 ms. The red line shows the time slice and the green lines show different Gaussians, the sum of which fits the red line.

https://doi.org/10.1371/journal.pone.0062234.g002

The Gabor wavelet produces a blurred impulse response around given frequency responses at each point in time. The blurring from a Gabor wavelet is in the form of a Gaussian [41], as illustrated by a time slice at time point 1460 ms shown in figure 2c taken from the scalogram in figure 2a. Further techniques have to be applied to the transformed data in order to extract the instantaneous frequency information. Standard ridge and skeleton methods do not perform well when there are many components, some of which remain very close for a while and separate again, or when they can die out, or when new ones can appear from nowhere [42]. As can be seen by the scalogram in figure 2a the data in the work presented here is of this type. Drawing upon the Gaussian nature of the impulse response from the Gabor wavelet, we apply a technique of fitting a sum-of-Gaussians model to the transformed data at each point in time [41]. Figure 2c shows such a fitting. Identifying the means and magnitudes of the means of the fitted Gaussians gives the instantaneous frequencies and their amplitudes respectively.

The next stage in preprocessing the data requires forming a time series of the instantaneous frequencies as they fluctuate over time, what we call a strand. These fluctuating frequency responses may also be intermittent due to the frequency response dropping out and starting again. Zero values are substituted into the time series strands during the drop out moments to identify the fact that there is no frequency response at those times. There may be many coexisting frequencies for each neural oscillator at each time point in a simulation, and therefore many coexisting strands. To obtain these strands, after the instantaneous frequencies at each point in time for an oscillator are calculated, the movement of each frequency is tracked over time so as to link them together into a single time series fragment.

The algorithm for forming the frequency time series strands has three parts. The first part is simply to sequence the nearest frequencies in time into a strand as follows:

T = start_time.

while T is not equal to end_time.

 For each unassigned instantaneous frequency at time point T create a new strand containing that frequency.

 T = T+1.

 while there are strands and frequencies within the distance limit L.

 Find the strand at time point T-1 with the closest frequency to one of the instantaneous frequencies at time point T.

 If the frequency is within limit L add the frequency to the strand and remove the strand from further consideration until the next iteration.

 end

end

Further to this, we need to cope with bifurcations in the oscillator behaviour when an oscillator in a particular state A1, flips to another state B1, and then flips again to a state A2 such that states A1 and A2 have the same number of coexisting frequencies and these frequencies have approximately the same values. Hence the system is returning to its original state (A1) after the middle state B1. In each state there may be several coexistent frequency strands. We wish the strands in the original state A1 and its return state A2 after the middle state, to be stitched together so as to maximize the strand length and as a result the correlation. The distance between the frequencies in the strands in the state A1 and state A2 may be near enough within a limit L to make a direct match as in the previous algorithm, due to the fact that there is a close continuation between the two. However, there are situations in which the values of the frequencies in A2 are not near enough for direct matching, but instead have values similar to how state A1 would have been at that time if the bifurcations had not occurred and state A1 had instead continued developing. That is to say, whilst being the same state as A1, state A2 is in a later stage of development. In such a situation we use regression to project where the frequencies of strands in the original state A1 would have progressed to, and match these projections to the frequencies of the strands in A2. We apply a maximum frequency distance limit L as before on this projected matching.

In order to stitch states in this way, we first group all strands together which share the same start time so as to identify them as being in the same state. The state matching algorithm then preferentially matches states nearest to each other whose strands have the closest frequencies or projected frequencies. There is a maximum time limit between states for which we allow such stitching to occur. In order to get the best matching between states, we first perform the algorithm with the constraint that stitched states must contain the same number of strands, and then perform the algorithm again without this constraint.

After the state stitching has been carried out we extract the individual strands that are contained within the states, as we only consider pairs of individual strands during correlation. The strands are time series of fluctuating frequencies sequenced by closeness. Each strand will have a start and end and may contain zero values in its time series where the frequency dropped out due to a bifurcation.

Mean Intermittent frequency correlation

For each pair of oscillators, m and n, there is a collection of fluctuating frequency strands scattered over the frequency domain and stretching over time. We correlate each strand in oscillator m with each strand in oscillator n, for all oscillator combinations in the network. We do this by passing a 100 millisecond window over time in incremental steps of 1 millisecond. In each of these windows we take the time series data in that window for all pairs of strands i and j, where i and j are from different oscillators m and n respectively. For each pair of windowed strands we then remove time points from each strand where both strands do not have a response at that time point in the window, or when both the frequencies in the strands at one time point are the same as at the previous time point. This results in two time series strands for the window at time t, w_m,i(t) and w_n,j (t). Both are the same length and are potentially shorter than the window size. Each time series contains only data where both original strands have a frequency response and they are both fluctuating. We then correlate these two series. We select only correlations where the coefficient is greater than or equal to 0.5 or the coefficient of the anti-correlation is less than or equal to −0.5, and the p-value for either is less than 0.05. By randomizing the order of one of the series and performing the same correlation and selection process we obtain a phantom correlation. We use phantom correlations to evaluate the importance of the measure of real correlation found. For both types of correlation we calculate the mean intermittent frequency correlation as follow:

Where n and m are oscillators, I(t) and J(t) are the total number of strands in the window at time t for each oscillator respectively, and i and j are particular strands within each oscillator. coef^* defines the value of a significant correlation coefficient as previously described. W(t) is the length of the two series w_m,i(t) and w_n,j (t), that only contain time points that have a fluctuating frequency response in both original windowed strands. 100 is the length of the window. Thus the significant correlation coefficient is normalized according to the length of the two series in that window. t is the time of the particular window and t_max is the length of the simulation time. The metric calculates all pairwise significant frequency correlations between all oscillators, normalizes them by their length, and averages them over time.

Synchronisation metric

All the simulations in this work consisted of 10 neural oscillators connected together. Each neural oscillator consisted of an excitatory layer and an inhibitory layer. We only calculated synchrony for the excitatory neuron layers in the oscillators. The spikes of each neuron in each excitatory layer were binned over time, and then a Gaussian smoothing filter was passed over the binned data to produce a continuous time varying signal. Following this, we performed a Hilbert transform on the mean-centred filtered signal in order to identify its phase. No band-pass filtering was performed during this process. The synchrony at time t was then calculated as follows:where θ_j(t) is the phase at time t of oscillatory population j. i is the square root of -1. N is the number of oscillators, and t_max is the length of time of the simulation.

Coalition entropy

Coalition entropy measures the variety of metastable states entered by a system of oscillators. We only calculated coalition entropy for the excitatory neuron layers in the oscillators. As with the synchrony metric, we calculated the phase of each oscillator at each time point t using a Hilbert transform. We then performed clustering at each time point by picking the two most synchronous oscillators/coalitions using the first equation defined for the synchrony metric. Once a pair was identified they were joined to form a new coalition and the new coalitions mean complex exponential phase was calculated for use in the future most synchronous pair selection process. A threshold of 0.05 from full synchrony was used to limit the cluster merging. The process was repeated until no oscillators/coalitions fell within the threshold to allow merging into a new coalition.

Having identified the synchronous coalitions at each point in time we calculated the probability p(s) of each coalition occurring from the number of times it appeared throughout the simulation. The coalition entropy H_c was then calculated as follows:where |S| is the number of possible coalitions given the number of oscillators in the system.

Hardware acceleration

Each of our simulations required 10 neural PING nodes each of 250 neurons, resulting in 2500 neurons and ≈880,000 synapses, and entailing an immense computational burden across the entire parameter space sweep in our experiments. To cope with this, we used the NeMo neural network simulator, which processes neurons concurrently on general purpose graphics processing units (GPUs) [43]. The NeMo software permits the addition of user plugins for neural models, which allowed us to implement both QIF and HH models for the NeMo simulator facilitating the work presented here.