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Effects of Correlated Input on Development of Structure in an Activity Dependent Network Alexander J. H. Fedorec CoMPLEX, University College London Supervisors: Dr. Simon Farmer, Dr. Luc Berthouze March 6, 2014 Abstract The complexity of the brain is much discussed both within the scientific community and amongst the public. The network of neurons that make up the brain are connected in a manner that is important to its correct functioning. The development of these connection, from a disorderd, unconnected group of neurons to a modularised, “small-world” network is activity-dependent. Data from the EEGs of pre term babies shows that the activity present in the developing brain exhibits long-range temporal correlations. We build a neuronal network model to asses whether the form of the activity input in to a developing network effects the topology of the developed network. Our results disagree with the a recent study that showed a difference between a network generated with random process input and one with LRTC input. We leave the door open for future investigation with a model such as ours. Chapter 1 Introduction The brain is a complex system, often referred to as the most complex system in the universe, which allows us to comprehend the world around us. How the brain carries out the complex and varied tasks that it is capable of remains an area shrouded in uncertainty. However, it is believed that the form of the brain, the layout of its constituent parts, is intrinsically linked to its function. The developing brain The developed brain consists of large numbers of neurons, clustered in to modules which carry out different functions (Chialvo 2004). The main process that occurs in the early stages of development is connectivity formation (Kostovic & Rakic 1990). Neurites grow out from the neuron, forming connections with the neurites of other neurons. This growth and connection is dependent on the levels of activity within the neuron and the activity of neighbouring neurons. This activity dependent connection formation was theorised by Hebb and can be summarised with the statement: “cells that fire together, wire together”. The developed brain Investigations into the connectivity of the brains of a number of species have shown that the network topology is “neither entirely random nor entirely regular” (Sporns & Kötter 2004). These neuronal networks are characterised by high levels of clustering and a small average path length (Sporns & Kötter 2004). Such networks are known as small-world networks and are seen, not just in the brain, but in many areas including the western United States electrical power grid and the network of collaborations of feature film actors (Watts & Strogatz 1998). As shown in figure 1.1, the topology of a small-world network falls between a regular network, in which nodes only form connections with their neighbours, and a random network in which nodes form connections that are not related to path length. 1 CHAPTER 1. 2 a) b) Figure 1.1: a) One can create a random network by rewiring a regular network, in which each node is connected to its nearest neighbours (in this case its four nearest neighbours). The rewiring is done by proceeding around the ring and at each node, with probability p, reconnecting the edge to its nearest neighbour to a node chosen uniformly at random. This is repeated for the second nearest neighbour of each node and so on until each edge in the original network has been considered. b) The normalised clustering coefficient, C, and mean path length, L, plotted against the rewiring probability. Where there is a small probability of rewiring, in the network produced some nodes will have long range connections which shortens the mean path length while maintaining a high level of clustering i.e. a small-world network. Figures from Watts & Strogatz (1998). What drives cortical development? In the early stages of development of the cerebral cortex, an “axonal scaffold” is formed by the subplate, bridging between the thalamus and the developing cortical plate (McConnell et al. 1989). The subplate is a transient structure which disappears after the first postnatal week (Price et al. 1997), though some cells may remain (McConnell et al. 1989). It is thought that the activity passed into the cortical plate by the subplate influences the development of the neuronal network of the cortex (Dupont et al. 2006). The subplate activity is able to provide this input due to its position as a thalamic intermediary (Dupont et al. 2006). Activity-dependent development Most research conducted in the area of connectivity development in the brain, a network in which development is activity-dependent, has focused on changes caused by suppressing or removing activity (Tolner et al. 2012, Dupont et al. 2006). These studies show reduced cortical patterning and weaker thalomocortical connectivity. A less well explored area is whether the form of the activity driving the development is important in the networks developed. The EEG activity measured from brains shows that the activity is discontinuous, a series of bursts of activity punctuating a constant low level of oscillation, see figure 1.2. Recent studies of EEG data in pre term babies has shown that the activity displays long-range temporal correlations (LRTCs) (Hartley et al. 2012). CHAPTER 1. 3 Figure 1.2: The EEG of a pre term baby shows low level noise interspersed with bursts of activity. Long-range temporal correlations A process with long-range temporal correlations is a process in which the autocorrelations decay slowly, typically with a power law like decay (Craigmile 2003). This is as opposed to a shortrange process in which the coupling of values decays rapidly the further apart they are in time (or in space). One can establish whether a process shows LRTCs by estimating its Hurst coefficient, H, with “ 12 < H < 1 corresponding to long-term dependence” (Davies & Harte 1987). Although it is beyond the scope of this report, it should be noted that estimating the Hurst coefficient should be done using more than one method as there are biases inherent in each technique (Dieker 2004). LRTCs and neuronal development Several papers have been written regarding the use of Poisson processes as an input to neuronal models (Liu et al. 2003, Brown et al. 1999). To our knowledge, the only research conducted to determine what impact this long-memory process has on connectivity development has been undertaken by Hartley (2014). As well as looking at the effect of an LRTC process on network topology, Hartley examines whether the dynamics of developed networks exhibit LRTCs. The results showed that small-world, modular and random networks can produce dynamics with long-range correlations. In a study of pre term baby EEG data, (Hartley et al. 2012) revealed that despite changes in connectivity during the developmental period, EEGs show discontinuous activity with the same Hurst exponent. Chapter 2 Replication of Activity-Dependent Neuronal Network Model In order to explore the effects of an LRTC process on the clustering motifs formed in a neuronal network, we use a model from Van Ooyen & Van Pelt (1996). This model of activity-dependent neurite outgrowth allows us to input various processes and analyse the network of connections that forms. In their model, two properties of the neurons interact with each other. One is the slow process of neuritic growth, leading to the formation of connection with “overlapping” neurons. The second is the fast dynamics of electrical activity. Intracellular calcium is intrinsically linked to neurite growth (Kater & Mills 1991). There is an optimum level of intracellular calcium, above and below which neurite outgrowth is inhibited and even reversed (Kater et al. 1988). Levels of intracellular calcium are altered through several mechanisms, such as membrane depolarisation, and have been implicated in controlling the morphology of neurons (Kater et al. 1989) as well as the formation of patterns in neuronal circuitry (Lipton & Kater 1989). As such, the model uses membrane potential, and the mean firing rate associated with that potential, to modulate the growth or retraction of the “neuritic field”. The membrane potential, Xi , of a cell i is described by the following modification of the shunting model (Van Ooyen & Van Pelt 1996): N X dXi = −Xi + (1 − Xi ) Wij F (Xj ) dT (2.1) j=1 where N is the total number of cells in the network, T is the membrane time constant, Wij is the connection strength between neurons i and j and F (X) = 1 1+ 4 e(θ−X)α (2.2) CHAPTER 2. 5 where F (X) is the mean firing rate and α and θ determine the steepness and the firing threshold respectively. The connection strength of two neurons is proportional to the area of overlap of their neuritic fields: Wij = Aij S (2.3) where Aij is the area of overlap which we consider as analogous to the total number of synapses formed between i and j. S is a constant representing the average strength of the synapses. Van Ooyen & Van Pelt (1996) suggest that calculating the actual area of overlap between neuritic fields is not necessary to capture the essential behaviour and that simpler functions can be used. To describe the growth of a neuron, which is dependent on its electrical activity, we take the change in the radius of the neurons neuritic field to be: dRi = ρG(F (Xi )) dT (2.4) where Ri is the radius of the neuritic field of neuron i and ρ is a constant determining the rate of growth. The function G can be any function that fulfils the following criteria: G(u) > 0 for u < i for u > i G(u) < 0 (2.5) for u = i G(u) = 0 This captures the property of neurons, described earlier, that there is a level of intracellular calcium, , beyond which growth will stop and further beyond which the neuron will retract. We use a function for G suggested in Van Ooyen & Van Pelt (1996). G(F (Xi )) = 1 − 2 1 + e(i −F (Xi ))/β (2.6) where β is a constant determining the steepness of the function. We used MatLab’s built in Runge-Kutta ordinary differential equation solver to produce solutions to the model described above. Where given, we used the parameter values set by Van Ooyen & Van Pelt (1996). Though the model set out by Van Ooyen & Van Pelt (1996) does not state that the plane which the neurons inhabit is toroidal, we have made it so. This allows us, when producing solutions where all neurons are identical other than in their position on a grid, to use only a few neurons and avoid boundary conditions. The neuritic growth/retraction threshold In order to look at the impact of changing , we placed 16 neurons on a grid, equally spaced and with their neuritic fields touching but not overlapping. Figure 2.1 shows the mean membrane potential and mean radius of the neurons over 30,000 timesteps. Increasing , increases the membrane potential threshold at which the CHAPTER 2. 6 neuritic field of a neuron begins to retract. By increasing this value we are able to alter the steady state membrane potential. A value too low leads to persistent oscillations and a network that never settles. The synaptic strength S We now explore the impact of changing the value of the synaptic strength parameter, S, in equation 2.3. A value of = 0.6 is used for these simulations as it produces a steady state but is, potentially, not incapable of being pushed away from it by small perturbations. Increasing S means that the overlap of neuritic fields has a greater impact on the change in a neuron’s membrane potential, X. Our simulations, shown in figure 2.2, exhibit behaviours that one would expect. As S increases, the membrane potential of a neuron is affected to a greater extent by the activity of its neighbours. This means that, for larger values of S, a small overlap with neighbours is able to generate a change in membrane potential that for lower values of S would require a larger overlap. This means that the maximum radius of the neuritic fields is reduced as S increases. Further, the stability of the steady state is reduced as the impact of small changes in overlap lead to larger changes in membrane potential. For later simulations we use S = 0.5 as this balances the need to allow neuritic fields to grow and form multiple connections while generating a network that isn’t so stable that external inputs are unable to produce any change in the system. CHAPTER 2. mean membrane potential 7 mean neuritic field radius mean R vs. mean X a b c d e f Figure 2.1: Time, along the x-axis of the first two columns, runs from 0 to 3 × 104 . The third column displays mean neuritic field radius along the x-axis against mean membrane potential on the y-axis. Sixteen identical neurons on a toroidal grid with initial neuritic fields touching but not overlapping. The synaptic strength S = 0.5. a) With = 0.05 the radii of the neurons grows slowly, meaning a slow growth in neuritic field overlap and as such slow increase in membrane potential. As epsilon grows [b) = 0.2, c) = 0.4] the rate of increase in radii increases. d) When = 0.6, following an overshoot, a steady state is reached. This occurs when e) = 0.8 but when f) = 0.97, the relaxation of the neuritic fields is so slow that a steady state is not reached before the simulation elapses. CHAPTER 2. mean membrane potential 8 mean neuritic field radius mean R vs. mean X a b c d Figure 2.2: As the synaptic strength, S, increases [a) S = 0.2, b) S = 0.5, c) S = 0.8, d) S = 1] the maximum radii that the neurons reach is reduced. Further, the stability of the steady state decreases with increasing S as a slight increase in neuritic field overlap has a greater impact on membrane potential change. Chapter 3 The Impact of External Input on Neurite Growth The only stimulation supplied to the neurons as described in the model so far, comes from interactions with the neuritic field of other neurons. We wish to determine the impact of external stimulation on the network of neurons. In order to do so we add an input to equation 2.1: N X dXi = −Xi + (1 − Xi )(Ii + Wij F (Xj )) dT (3.1) j=1 where Ii is an input to neuron i. Van Ooyen & Van Pelt (1996) describe the impact of a constant input however we are interested in discontinuous inputs. Figure 3.1 shows the decreasing importance of the neuritic fields of other neurons as the strength of a constant input into a neuron increases. With a very high input strength the neuritic fields are barely able to grow, limiting the potential connectivity of the network. This is obviously not useful when considering connection formation and shows the necessity of restricting input strength. 3.1 Discontinuous Input Electrical activity in the brain does not take the form of a constant input of the form used in figure 3.1. Nor is it a continuous oscillation. EEGs show low level noise interspersed with bursts of activity (André et al. 2010). Anderson et al. (1985) carried out analysis on EEG readouts from pre-term babies. Their results show that the average duration of a burst of activity is 4-5 seconds. This value doesn’t show variation with conceptional age over the range that tested. Inter-burst intervals show a decrease both in average and longest period as conceptional age increases, with values of 7-12 seconds and 9 CHAPTER 3. mean membrane potential 10 mean neuritic field radius mean R vs. mean X a b c d Figure 3.1: With a constant input to each neuron the membrane potential is no longer solely effected by overlap with neighbours. As the strength of the input increases, the influence of the neighbours decreases. The neurons have = 0.6 and the synaptic strength = 0.5. CHAPTER 3. 11 20-48 seconds respectively. More recently, a study by Hartley et al. (2012) quotes a figure of 251.7 ± 55.1 events per hour, or an average inter-event interval 14 seconds. They show no variation due to conceptional age but do note that babies with cerebral haemorrhages have ”significantly lower event rate” of 110.1±38.9 events per hour. The EEG recordings used in the article from Hartley et al. (2012) were taken over a longer period (median duration 21.6 hours) than the Anderson et al. (1985) study (26.5 minutes), though the former only had 11 subjects compared to the latter’s 33 subjects. 3.1.1 Poisson Process Algorithms to generate Poisson processes are well documented. We use an algorithm described by Pasupathy (2011). In order to show the effect of such a process on the model, as described, we generated a Poisson process with a mean rate of 0.07 events per second, corresponding to the figure of 251.7±55.1 events per hour measured by Hartley et al. (2012). The duration for each event was taken to be 5 seconds, as per the findings of Anderson et al. (1985). Figure 3.2 shows the effect that such a process has on the same network of neurons on a grid that we have used previously. As the strength of the input increases, its impact on the activity of a neuron overtakes that of the neuron’s neighbours. This leads to an increase in the rate of change of membrane potential and neuritic field radius. If there is a steady state for a particular set of neuron parameters, the input can push the neuron away from it. Again we see that when the strength of the input is high, the maximum radii of the neurons is limited. As such, we decide to increase the value of that we use for our simulations to = 0.8. This should allow the network to avoid falling in to an endlessly oscillatory state. Random spatial positioning Now that we have established the ability to input a process in to a network of neurons we can begin to look at the effect of such processes on the formation of connections. While on a grid, each neuron can form connections with its neighbours in the North, South, East and West positions and may, if the value for the synaptic strength is suitably low, form connections beyond. This is obviously very restrictive to the characteristics of the network that can form. As such, the initial positions of the neurons were randomised1 , still on a toroidal plane with the same density of neurons as before. First we tested whether the number of neurons in the network made a noticeable difference to network dynamics. Figure 3.3 shows that changing from 49 to 100 neurons leads to a slight dampening in the rate of increase in mean membrane potential but the ”steady” level of the membrane potential and mean radii reached were the 1 The seed used when generating the positions was the same, unless stated, in order to avoid the outcomes differing due to differences in initial set up rather than differences due to variables we control. CHAPTER 3. mean membrane potential 12 mean neuritic field radius mean R vs. mean X a b c d Figure 3.2: A Poisson process input with mean rate of 0.07 and duration of events equal to 5. The neurons have = 0.6 and the synaptic strength = 0.5 and are equally spaced on a toroidal plane. Increasing the strength of the input [a) 0, b) 0.1, c) 0.2 and d) 0.5], destabilises the network due to the ability of the input to push the membrane potential of the neurons over its value, thus leading to neuritic field retraction. As the strength of input increases, the neurons’ membrane potentials are increasingly linked to the fluctuations of the input. CHAPTER 3. a) = 0.8, N = 49 13 b) = 0.8, N = 100 c) = 0.7, N = 100 Figure 3.3: As the number of neurons in the network is increased from a) 49 to b) 100, the dynamics remain rather similar other than a slight dampening in the rate of increase in mean membrane potential (meanX). A bigger difference is clearly seen in the change of from b) 0.8 to c) 0.7. As we saw in previous sections, reducing reduces the stability of a steady state for membrane potential. same. One can also see that by lowering the value of from 0.8 to 0.7, the input is able to perturb the dynamics of the network enough to cause a dramatic drop in mean membrane potential and an associated change in neuritic field growth. In later simulations we run with both = 0.8 and 0.7 in order to ascertain whether periods of instability are necessary to generate different network outcomes from different inputs. Another important result from setting equal to 0.7 is that it shows that we can’t assume a system will remain in a steady state just because it has been in one for a certain period. If we had cut off the simulation at time-step 20000, the = 0.8 and 0.7 would have looked much the same. 3.1.2 LRTC Process We have shown how a Poisson process as an input to a neuronal network can affect the dynamics of that network. What we are really interested in, however, is whether an LRTC process produces different network characteristics to the Poisson process. Our LRTC process is a fractional, autoregressive integrated moving-average (FARIMA) process. This works by allowing the degree of differencing to take fractional values, with a differencing value between 0 and 21 producing a long-range dependence (Hosking 1981). As stated previously, an LRTC can be described by its Hurst exponent. Hartley et al. (2012) found that the Hurst exponent of EEGs in pre-term babies was ∼ 0.6 − 0.7. We created LRTC processes with the same inter-event intervals as the Poisson processes used in the previous section and used them as the input in neuronal networks with the same initial conditions as used in figure 3.3. The results, in figure 3.4 show similar dynamics to those produced CHAPTER 3. a) = 0.8, N = 49 14 b) = 0.8, N = 100 c) = 0.7, N = 100 Figure 3.4: With an LRTC process as input, the neuronal dynamics appear very similar to those produced by a Poisson process with identical strength, mean interevent interval and event duration. using a Poisson process input. 3.2 Network Topology Although the neuronal dynamics of the networks with Poisson and LRTC process input look similar, this tells us little regarding the topology of the respective networks. In order to asses whether LRTC process input creates networks with different topologies to memoryless processes we look at the clustering coefficient and mean path length of the networks. Figure 3.5 shows how these two statistics vary as the networks develop with the two different processes as input. The results show very similar trajectories for clustering coefficient and mean path length. This would seem to indicate that the networks connectivities are very similar and the long-term memory process doesn’t generate a different form of network to that of a memoryless process. However, in order to make a genuine comparison between the to simulations the clustering coefficient and mean path length must be normalised. From each of the connectivity matrices at the final step of the networks generated, we created 100 randomised networks. This was done by randomly shuffling the the values in the upper triangle of the connectivity matrix and then reflecting along the diagonal to produce a symmetric matrix2 . This method retains the number of connections and the weight of connections within the network. The mean clustering coefficient and mean path length of these random networks allow us to normalise the mean clustering coefficient and mean path length from the the simulated networks (Bassett & 2 2.3 Our connection matrix is symmetric because the weight of Aij = Aji , see equation CHAPTER 3. 15 a b Figure 3.5: The mean clustering coefficient and mean path length with LRTC process input (red) are very similar to those of the network with Poisson process input (blue). The only noticeable differences are an earlier second spike in clustering coefficient with an LRTC process with = 0.7 (a) whereas the Poisson process with the same network set up takes longer to affect a second spike in clustering coefficient but a third spike follows soon after. Further, with = 0.8 (b) the drops in clustering coefficient with an LRTC input lag behind the same network with a Poisson input. CHAPTER 3. 16 Bullmore 2006). Table 3.1 shows these normalised values for the last point in the time series. This also shows that the networks generated by both inputs share similar topologies. However, we want to see if the topologies remain as synchronised throughout the time series as figure 3.5 would suggest. Due to time restraints we were unable to normalise the clustering coefficient and path length at each data point but we were able to do so at a less precise level (1 normalised point for every 100 data points for LRTC and 1 for every 500 for Poisson). Figure 3.6 shows the normalised values. Although the clustering coefficient for LRTC input has a greater level of variation on short time-scales, the Poisson process follows the same trend as the LRTC process. For the mean path length the similarity is much clearer. Poisson LRTC Mean Clustering Coefficient measured random normalised 0.2431 0.0264 9.2083 0.2420 0.0247 9.7976 Mean Path Length measured random normalised 5.9710 3.6081 1.6549 6.0478 3.6524 1.6558 Table 3.1: The mean clustering coefficient and mean path length of the networks of 100 neurons, with = 0.8. These are taken from the connectivity matrix at the last recorded time step. We can calculate the small-world index from the normalised values shown in table 3.1. For the LRTC network the small-world index is 9.7976/1.6558 = 5.9171 and for the Poisson network it is 9.2083/1.6549 = 5.5643. A value of 1 would show that the network has similar characteristics to that of a random network (Bassett & Bullmore 2006). Both of these networks have a value greater than one, showing that they both display small-world network characteristics. In fact one can see from figure 3.6 that the network shows small-world properties throughout the time-series. CHAPTER 3. 17 a b Figure 3.6: The a) normalised mean clustering coefficient and b) normalised mean path length with LRTC process input (red) and Poisson process input (blue) with = 0.8. The LRTC line is calculated to a greater precision (1 point at every 100th output point [Not every 100th unit time-step. Due to using a Runge-Kutta method, output point are not equally spaced or corresponding to integer timesteps.]) compared to the Poisson line (1 point every 500th output point). One can see that although there is a greater degree of variation in the LRTC line, the Poisson process does follow the same overall trend in clustering coefficient. The similarity is much more noticeable for the mean path lengths. It should be noted that, at all points in the time series, both networks display small world properties i.e. clustering coefficient / path length > 1. Chapter 4 Discussion We replicated a model of neuronal dynamics described by Van Ooyen & Van Pelt (1996) and extended it to allow for discontinuous input. Difficulties arose while trying to run simulations with this model due to the difficulties of integrating discontinuous functions in MatLab. We adopted a ‘hybrid’ method by using the Runge-Kutta method of analysis between events. This allowed us to take advantage of the efficiency savings that Runge-Kutta offers while also being certain that the events weren’t missed. Though this method was faster than simply reducing the size of integration time steps to a suitably small number, the computation still took 1-3 hours with 100 neurons. As is evidenced by the graphs in figure 3.3 and figure 3.3 in which = 0.7, extending the simulation time is necessary in order to capture behaviours that we are likely missing. These graphs also bring in to question whether the membrane potential and radii with values of = 0.8 are as stable as we assume. The methods for calculating clustering coefficients and mean path lengths were taken from the Brain Connectivity Toolbox (Rubinov & Sporns 2010). Bolaños et al. (2013) point out certain errors in a number of methods for calculating clustering coefficients in weighted directed networks, mentioning Rubinov & Sporns (2010) in the paper. A new method is proposed and should be used in future work when calculating network topology characteristics. The graphs in figure 3.3 and figure 3.4 in which = 0.7 show activity occurring at the end of the simulation. It is necessary to run the simulations for longer in order to ascertain whether the membrane potential finds a steady state or whether oscillations continue. In addition to running for longer, it is also necessary to use more neurons. The large amount of variation in the normalised clustering coefficient graph in figure 3.6 is likely to be due to the disproportionate effect that the disconnecting/reconnecting of a small number of neurons can have on a network with a relatively small number of neurons in it. 18 CHAPTER 4. 19 Our results currently show no difference in the connectivity of neuronal networks when stimulated with LRTC or Poisson process input. This goes against the findings of Hartley (2014), who showed that there was both a difference in the way connectivity evolved and in the final topology. The difference in findings could be due to several factors but is most likely down to the parametrisation of the model. We made no effort to develop neurons that reflected the “physiology” of those that Hartley (2014) modelled. The strength and relative frequency of the input processes was not reflective of the processes used by Hartley (2014) either. Since there was no difference in the network between the two inputs, it may be the case that the impact of the inputs on the network wasn’t great enough. We chose parameters for the network in order to try and generate one that was reasonably stable. Although we attempted to determine the effect on a slightly less stable neuronal set up, this simulation wasn’t run for a long enough period to be able to assess the effect of this change. In the future it would be of interest to determine whether increasing the strength of the input on both a relatively stable and relatively unstable network produces greater differentiation between the two processes. Another alteration that would be interesting to explore would be to change to frequency of the input. The values we used of 0.07 for the mean rate and 5 for event duration were taken from literature. However, in the literature these were in relation to a real clock. In our simulation we had them in relation to unit simulation time. We did not assess the relation of a unit of simulation time to that of a second. This relation can be altered by varying ρ, the growth rate, in equation 2.4 (Van Ooyen & Van Pelt 1996). Chapter 5 Conclusion The brain is a complex network of neurons and their connections. The topology of the neuronal network is important for the proper functioning of the brain. It is thought that connectivity development in the brain is activitydependent. Recent EEG data from pre term babies has shown that activity in the developing brain shows long-range temporal correlations (Hartley et al. 2012). The question of whether the form of the activity input to the developing brain has an impact on the developed network topology has only recently been addressed by Hartley (2014). We built an activity-dependent neuronal network model to determine whether the topology of a network generated under the influence of an LRTC process differs from that formed under a Poisson process. 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