Do Brain Networks Evolve by Maximizing Their Information Flow Capacity?
[['A plethora of phenomena in nature can be effectively described by networks .', 'Neuroscientists have used tools for the analysis of complex networks that help realize even more deeply the functionality and structure of the brain .', 'It was found that many aspects of brain network structures are typical of a wide range of non-neural or non-biological complex networks [1 , 2] .', 'One of the main findings in neuroscience is the modular organization of the brain , which in turn implies an inherent parallel nature of brain computations [1] .', 'Modular processors have to be sufficiently isolated and dynamically differentiated to achieve independent computations , but also globally connected to be integrated in coherent functions [1] .', 'It has been revealed that the cortical network is a hierarchical and clustered network with a complex connectivity [3] .', 'A possible network description for this modular organization is that brain networks may be small-world structured [4] with properties similar to many other complex networks [5] .', 'This viewpoint has been driven by the systematic finding of small-world topology in a wide range of human brain networks derived from structural [4] , functional [6] , and diffusion tensor MRI [7] studies .', 'Small-world topology has also been identified at the cellular-network scale in functional cortical neural circuits in mammals [8] and also in the nervous system of the nematode Caenorhabditis elegans ( C . elegans ) [9] .', 'Moreover , this topology seems to be relevant for the brain function because it is affected by diseases [10] , normal ageing , and by pharmacological blockade of dopamine neurotransmission [11] .', 'Synchronization is ubiquitous in nature .', 'Insightful findings regarding synchronization in complex networks were reviewed recently in Ref .', '[12] .', 'Recently , synchronization in complex modular or clustered networks has been investigated [13 , 14] .', 'It appears as the interplay between the intrinsic dynamics associated to the nodes of the network and its graph topology and connecting functions .', 'In this work , synchronization will be considered as a mean to quantify functional behaviors of the brain dynamical networks ( BDNs ) studied .', 'By BDN we mean a network that represents the connectome equipped with neural dynamics for its nodes to account for their time evolution .', 'Mathematical and computational approaches have a long tradition traced back to the early mathematical theories of perception [15] and of current integration by a neural cell membrane [16] .', 'Hebb’s idea on assembly formation [17] inspired simulations on the largest computers available at that time ( 1956 ) [18] to understand the relation between neural connectivity strength and response , i . e . behavior .', 'It is a learning rule which proposes an explanation for the adaptation of neurons during the learning process .', 'The simultaneous activation of pairs of neurons leads to pronounced increases in their synaptic strength .', 'There is also the possibility of many other kinds of learning [19] .', 'In this work we find evidence of Hebbian-like and no Hebbian-like learning rules characterized by the relationship between the addition of synapses and the increase or decay in the synchronization behavior of neurons in the evolved BDNs .', 'Several brain models have been studied so far that do not take into account particular behaviors of isolated neurons , but higher level functions of the external stimulus and the behavioral response attributed to ensembles of neurons of cortical areas .', 'There are two classes: Those based on collective functional dynamics of local groups of neurons , such as the Wilson-Cowan model for the cortical and thalamic nervous tissue [20] , and those based on the conditional probabilities ( as well as information and mutual information ( MI ) ) of stimuli and responses , prominent examples of which are the Bayesian brain hypothesis [21] and the infomax theory [22] .', 'Recently , it was shown that most of the probabilistic brain models can be unified under a single free-energy principle [23] , the one that interprets the brain as a system that tries to minimize surprises of the sensations from the world .', 'If the brain learns by maximizing the MI between stimuli and response [22] , or by updating the internal model of probabilities by using Bayesian techniques [21] , or by minimizing the free-energy [23] , the surprise of the response provides little insight about the dynamical mechanisms appearing in a brain network when it evolves based on such principles .', 'The drawback however of the probabilistic brain models is that they describe little about the underlying dynamical structure of what is really happening in the neural level and the representation of the stimuli [24] .', 'For deterministic systems with correlations , an appropriate quantity for measuring the transfer of information is the Mutual Information Rate ( MIR ) , the MI per time unit .', 'In Ref .', '[25] , the authors have developed alternative methods to overcome problems that stem from the definition of probabilities from time series and derived an upper bound for the MIR , Ic , between two nodes of a complex dynamical network from time averages that do not rely on probabilities , but instead on the two largest Lyapunov exponents l1 , l2 of the subspace of the network formed by the two nodes ( see Eq ( 10 ) in Materials and Methods ) .', 'In our study , Ic stands for the upper bound for the information transferred per time unit between any two nodes of the BDN , what represents the information flow capacity of the BDN .', 'We discuss more on these in Materials and Methods , Section Upper Bound for MIR .', 'Inspired by the infomax theory and by theoretical studies that proposes that the maximization of information transmission between subsystems can be used as a principle for understanding the development and evolution of complex brain networks ( see Ref . [2] and references therein , and Ref . [26] ) and , aiming at elucidating the microscopic dynamic mechanisms associated to the evolution of brain networks , we propose a working hypothesis and , provide evidence that a brain dynamical network may evolve based on the maximization of the information flow capacity it can internally handle at each step of its evolution process , i . e . a new inter-neuron connection is established if it leads to a subsequent increase of the information flow capacity of the new brain circuitry .', 'Our hypothesis is based on the internal neural network dynamics and on a plausible model for brain structure per se without the need to resort to probabilistic models based on the external influence in the brain and its response .', 'We have been able to show that our evolved BDNs present similar synchronization and information flow capacity behaviors with those found for the simulated dynamical networks for the brain structure of the C . elegans and humans .', 'Moreover , we show that BDNs evolved with coupling strengths that maximize the information flow capacity are the ones with the smallest spectral graph distance from the BDNs of the C . elegans and humans , and that , during the growing process , their information flow capacity increase is related to moderately low amounts of global neural synchronization .', 'This work provides ample evidence that brain networks may grow by maximizing the capacity of information flow they can internally handle , driven by global no Hebbian-like evolution processes , according to which the addition of interconnections between clusters during the evolution process leads to a decrease in the global synchronization level of the BDN .', 'This behavior is accompanied by a similar no Hebbian-like learning process for neurons in some of the clusters and by Hebbian-like processes for neurons in the remaining clusters , leading to an increase in the synchronization level between these neurons during brain network evolution .', 'Effectively , the no Hebbian-like mechanism is akin to the unlearning anti-Hebbian mechanism of Crick and Mitchison [27] that refers to the elimination of unnecessary connections to prevent overload and to render the network more efficient .', 'In our evolution we do not delete links .', 'However , both mechanisms lead to a decrease of synchronization and to more efficient networks , being in our case the evolved networks able to maximize their information flow capacity .', 'Global synchronization takes into consideration the synchronous behavior of all neurons in the BDN whereas local synchronization of neurons in a cluster of the BDNs .', 'Finally , our work shows that optimising information flow capacity leads to evolved networks that are heterogeneous , in accordance with the line of research in Ref .', '[2] , where the authors report on a mathematical model for the evolution of heterogeneous modules in the brain based on the maximization of bidirectional information flow transmission .'], ['We present in panels ( A ) , ( B ) of Fig 1 the global synchronization measure ρ and upper bound of information flow Ic for the BDN of the C . elegans in the parameter space of chemical coupling gn in [0 , 2] and electrical coupling gl in [0 , 2] .', 'We refer the reader to Materials and Methods , Subsection C . elegans Data .', 'A direct comparison between the two panels reveals a number of conclusions for the different parameter space regions .', 'At first , for relatively high chemical and electrical couplings almost full global synchronization can be achieved ( yellow and red regions in Fig 1 ( A ) ) .', 'For the same region , panel ( B ) shows an almost absence of capability of information transmission as the upper bound for MIR , Ic ≈ 0 ( dark blue region ) .', 'High levels of global synchronization accompanied by low values of Ic indicate that not only neural activities are similar but also they have very low entropy since both Lyapunov exponents λ1 , λ2 are practically zero leading to their difference Ic ≈ 0 ( see Materials and Methods , Subsection Upper Bound for MIR ) .', 'Second , for chemical couplings smaller than 0 . 3 ( i . e . gn ∈ [0 , 0 . 3] ) and electrical couplings gl ∈ [0 , 2] , we observe a multitude of different functional behaviors: There are regions of high synchronization ( red region in panel ( A ) ) and low Ic ( blue region in panel ( B ) ) and others with exactly the opposite behavior ( i . e . low global synchronization accompanied by high information flow capacity ) .', 'These different functional behaviors will become even more evident in Fig 2 , where we plot the regions between the left vertical axes and the white dotted lines of Fig 1 in a finer resolution .', 'Our findings suggest that for the C . elegans BDN , for most of the chemical and electrical couplings , global brain synchronization is roughly speaking inversely related to the information flow capacity , Ic .', 'As we shall see in Materials and Methods , Section Similarities between C . elegans and human BDNs for a more detailed analysis of the C . elegans and human BDNs , high synchronization as depicted by ρ implies small information flow capacity .', 'The relation between ρ and Ic can be better understood in the basis that typically ρ ∝ 1/λ2 , where λ2 is the second largest Lyapunov exponent of the BDN , for the so-called non-excitatory networks [28 , 29] .', 'The non-excitatory character is expected to be prominent when the electrical coupling has a dominant contribution to the behavior of the network with respect to the chemical , a situation that promotes global neural synchronization .', 'We have performed a similar study for the global synchronization ρ and upper bound of information flow Ic for the human brain networks in Materials and Methods , Subsection Human Subjects Data , shown in Fig 1 ( C ) and 1 ( D ) .', 'Particularly , we first prepared parameter spaces for all six human subject BDNs with the same coupling ranges with those of the first two panels of the C . elegans and then computed their average , presented in Fig 1 ( C ) and 1 ( D ) .', 'The averaged ρ and Ic quantities from the six subjects are indicated by ⟨ρ⟩6 and ⟨Ic⟩6 , respectively .', 'A direct comparison between panels ( C ) and ( D ) of Fig 1 for the humans reveals a number of parameter space regions associated to different functional behaviors , similar to those observed for the C . elegans .', 'For relatively high chemical and electrical couplings almost full global synchronization is achieved ( yellow and red regions in Fig 1 ( C ) ) since ⟨ρ⟩6', '≈ 1 . For the same region , Fig 1 ( D ) shows an almost absence of information flow capacity as ⟨Ic⟩6 ≈ 0 , the reason being the same as for the C . elegans BDN .', 'For chemical couplings smaller than 0 . 2 ( i . e . gn ∈ [0 , 0 . 2] ) and electrical couplings gl ∈ [0 , 2] we observe regions of high synchronization ( red regions in Fig 1 ( C ) ) and low ⟨Ic⟩6 ( blue region in Fig 1 ( D ) ) , as well as others with exactly the opposite property , having low synchronization and large ⟨Ic⟩6 .', 'In order to look deeper into the details of the functional behaviors of these BDNs and , to understand the structural and functional similarities between them , we study in Materials and Methods , Section Similarities between C . elegans and human BDNs , zoom-in plots of the previous parameter spaces .', 'There has been enormous research devoted on the C . elegans worm which has revealed its ability to learn about mechano , chemo and thermosensory inputs and stimuli [30 , 31] .', 'It was also shown that its neural system has the ability to distinguish between tastes , odours or any indication related to the presence or absence of food .', 'It also shows different kinds of learning behavior , such as associative ( classical conditioning and differential classical conditioning ) , and non-associative forms of learning , such as habituation and dishabituation [32] .', 'These properties are reminiscent of the human brain ability to adapt to different stimuli and environments .', 'In Fig 2 , we present a finer resolution version of the parameter spaces of Fig 1 for the C . elegans and humans .', 'They allow us to reveal the extraordinary similarity on the functional level , the global synchronization and Ic patterns between the C . elegans and human BDNs .', 'In these plots , wherever we observe high synchronization as evidenced by orange and red regions in panel ( A ) for the C . elegans and ( C ) for the human brain network , the upper bound for MIR , Ic , that stands for the internal information flow capacity of the BDN , is small ( blue region ) and vice-versa .', 'Since the C . elegans brain connectivity network is about four times smaller in size than the human BDNs [33] studied here , we should expect that the global synchronization and upper bound for MIR patterns could occur for different ranges of chemical and electrical couplings as compared to those of Fig 1 ( see for example Ref . [34] ) .', 'Therefore , a rescaling of the coupling strengths was employed to allow for both BDNs to have the possibility to produce equivalent dynamical behaviors .', 'This rescaling is described in Materials and Methods , Subsection Rescaling of Chemical and Electrical Couplings for Parameter Spaces of Networks with Different Eigenvalue Spectra .', 'It is worth noting that there is an optimal coupling range for both BDNs that allows for large information flow capacity in the brain networks ( orange and yellow regions in panels ( B ) , ( D ) of Fig 2 ) , a coupling range that promotes moderately low global synchronization !', 'In Materials and Methods , Subsection A Model for Brain Network Evolution Based on the Maximization of Information Flow Capacity , we propose an artificial brain network evolution model that presents important structural and functional properties of the BDNs of the C . elegans and humans .', 'It is based on the combined effect of chemical and electrical synapses and , on a topology reminiscent of interconnected brain communities found in these BDNs .', 'We use chemical synapses for the communication of neurons of different clusters ( inter-cluster connections ) and electrical for the communication of neurons within each cluster ( intra-cluster connections ) .', 'Here , we compare the functional properties of this brain network evolution model with those of the C . elegans and humans .', 'By functional we mean the properties of the dynamics of the BDN such as local , global synchronization and information flow capacity .', 'Particularly , we report on the similarities we found for the functional properties of the BDNs of the C . elegans and humans , and for those of the proposed model for brain network evolution .', 'To support further the validity of the presented results for the model for brain network evolution and to show its independence on the particular initial small-world cluster configuration , we computed its parameter spaces averaging over the functional measurements obtained from five different initial small-world cluster configurations ( for a discussion about the creation of small-world networks or clusters see Materials and Methods , Subsection Analysis of Networks and Communities ) .', 'In all cases , we used the same evolution process as described in Materials and Methods , Subsection A Model for Brain Network Evolution Based on the Maximization of Information Flow Capacity .', 'The results of this study are shown in Fig 3 , where ⟨ρ⟩5 stands for the average of the global synchronization of the evolved BDNs for the five realizations and ⟨mMIR⟩5 for the average maximal value of the MIR for the same realizations .', 'We first provide evidence in panel ( A ) and ( C ) to ( H ) that the finally evolved BDN of small-world clusters captures similar local and global synchronization properties to those of the C . elegans BDN .', 'The local synchronization ρci of the i-th community ( Fig 3 ( C ) –3 ( H ) ) also reproduces similarly the global synchronization patterns of Fig 2 ( A ) and 2 ( C ) for the C . elegans and human BDNs .', 'Comparing these panels , we conclude that the averaged synchronization measure ⟨ρ⟩5 of the brain network evolution model ( Fig 3 ( A ) ) attains almost similar values in the same coupling regions to those of the C . elegans in Fig 2 ( A ) .', 'The different ranges on the horizontal axes of the chemical coupling strength gn can be explained by the fact that the finally evolved network consists of 60 neurons whereas the C . elegans of 277 neurons ( for more details see Materials and Methods , Subsection Rescaling of Chemical and Electrical Couplings for Parameter Spaces of Networks with Different Eigenvalue Spectra ) .', 'It is remarkable that the finally evolved BDN exhibits almost identical synchronization and information flow capacity features as the BDN of the C . elegans and humans for almost all coupling ranges considered .', 'Particularly , focusing on panels ( A ) , ( B ) ( for the C . elegans ) and ( C ) , ( D ) ( for the humans ) of Fig 2 and , on ( A ) , ( B ) of Fig 3 ( for the brain network evolution model ) , we observe an almost identical pattern of global synchronization and information flow capacity as depicted by Ic and its averages .', 'Again , here we have used the rescaling in Materials and Methods , Subsection Rescaling of Chemical and Electrical Couplings for Parameter Spaces of Networks with Different Eigenvalue Spectra , to create networks that can potentially reproduce similar dynamical behaviors for the finally evolved BDNs , in agreement with those identified for the C . elegans and human BDNs earlier .', 'The upper bound for MIR , Ic , depends on various factors , such as network topology and connectivity patterns , coupling strengths and types ( chemical , electrical ) , synchronicity , etc .', 'The values of Ic , the information flow capacity , can be comparable ( or different ) for networks with different topologies .', 'This is due to the fact that Ic is a function of the chemical and electrical coupling strengths .', 'The surprising fact is that as we evolve a network with an initial small-world network configuration , by maximizing information flow capacity , the final network exhibits not only similar topological ( structural network characteristics ) but also similar functional or behavioral ( synchronization and upper bound for MIR ) properties as those found for the brain dynamical networks of the C . elegans and human subjects ( see Materials and Methods , Subsection Spectral Similarity of C . elegans and Human Brain Networks with those of the Model for Brain Network Evolution ) .', 'For completeness , in S2 Fig , we present a similar analysis to the one of Fig 3 based on Erdős-Rényi random networks ( panels ( A ) , ( B ) ) , scale-free ( Barabási-Albert ) ( panels ( C ) , ( D ) ) and star topologies perturbed by 20% for the clusters of the model for brain evolution of 60 neurons and 6 small-world clusters ( panels ( E ) , ( F ) ) and found out that the evolution model fails to capture similar functional ( local and global synchronization and , information flow capacity patterns in the parameter spaces ) as the same model equipped with small-world topologies for its clusters ( see Fig 3 ) .', 'Here , we are interested in studying and proposing a model for brain network evolution that is able to reproduce not only similar functional properties such as information flow capacity and , local and global synchronization properties , but also importantly to reproduce similar structural properties for the finally evolved full brain network and of its clusters .', 'Based on our results so far , we show next that the initial cluster configuration able to fulfil both requirements is the small-world cluster topology .', 'The discussion about the results of Fig 4 in Materials and Methods suggest that the evolution process of a basic small-world clustered network is capable of generating an evolved one with similar structural properties with those for the C . elegans and human BDNs ( see Materials and Methods , Subsection Structural Properties of the Model for Brain Network Evolution ) .', 'The structural similarity to the human brain network is even more remarkable if the couplings of the network to be evolved are within the range that promotes high levels of information flow capacity and low neural synchronization , a prominent example of which is case 𝓑 ( for a definition and discussion about cases 𝓐 , 𝓑 , see Materials and Methods , Subsection Brain Network Evolution Promotes Global no Hebbian-like and , Local Hebbian-like and no Hebbian-like Evolution Learning ) .', 'Here we study how close the normalized Laplacian spectral plots of the networks considered in Fig 4 ( Q ) are from those of evolved BDNs .', 'Particularly , we examined the spectral similarity by comparing spectral plots and computing their average Euclidean distance following Ref .', '[35] .', 'We refer the reader to Materials and Methods , Subsection Normalized Laplacian Spectra , for the details of our computations .', 'We compared the normalized Laplacian spectral plot of the C . elegans and of the six human brain networks with the spectral plots of all finally evolved networks of the five realizations used to compute the averaged parameter spaces of the model for brain network evolution of Fig 3 ( A ) and 3 ( B ) .', 'We provide additional support by demonstrating in the last four panels of Fig 5 results from a similar comparison between the C . elegans and humans with a double-sized version of 120 neurons of the model for brain network evolution of the Materials and Methods , Subsection A Model for Brain Network Evolution Based on the Maximization of Information Flow Capacity .', 'The normalized Laplacian spectra for the C . elegans and for the averaged over the six human BDNs are shown in Fig 4 ( Q ) .', 'As it can be seen , both spectra share some common features: Both show a left-skewed distribution in which the largest eigenvalue is closer to one in agreement with results in Ref .', '[35] .', 'Also , the distributions show peaks around one and the eigenvalues are scattered at the beginning of the spectra , suggesting similarities in their community structure , reminiscent of their small-worldness .', 'Although the spectral plots of both cases of the model for brain network evolution shown in Fig 4 ( R ) do not exhibit all properties of the spectral plots of the C . elegans and humans of Fig 4 ( Q ) , maybe due to the considerably smaller size of the former networks , they do exhibit interesting similarities: They are both left-skewed distributions with a peak around 1 . 3 , being closer to 1 than to', '2 . We also observe low relative frequency eigenvalues at the beginning of both spectra .', 'Both spectral properties suggest similarities in their community structure [35] as well ( i . e . their small-worldness ) .', 'This is in accordance with the spectral plots of the averaged human and C . elegans brain networks of Fig 4 ( Q ) .', 'We argue that this similarity comes from the small-worldness of the communities .', 'The close relation between the normalized Laplacian spectra of Fig 4 suggests the existence of common underlying structural properties of the neural networks of the C . elegans , the humans and the model for brain network evolution .', 'We measured the similarity between the spectral plots of the C . elegans and the averaged human brain network with the averaged model for brain network evolution and , plot in panels ( A ) , ( B ) of Fig 5 their spectral distance D for different chemical and electrical coupling ranges .', 'In this framework , the closer D is to zero , the closer structurally the compared networks are .', 'In particular , panel ( A ) is the parameter space for the spectral distance between the C . elegans and the averaged model for brain network evolution and , panel ( B ) is a similar plot for the spectral distance between the averaged human brain network and the same model for brain network evolution .', 'We pinpoint case 𝓐 by a ▲ and case 𝓑 by ● .', 'Such results allow one to draw interesting relations between structure and function of the proposed model for brain network evolution with the structural properties of the brain networks of C . elegans and humans .', 'Panels ( A ) and ( B ) of Fig 5 already reveals that the smallest mean spectral distance happens for the pair of chemical and electrical couplings that gives rise to BDNs that present small amount of synchronization and high information flow capacity , in other words to cases such as 𝓑 .', 'In contrast , one of the largest mean spectral distances was found for case 𝓐 that promotes high amount of neural synchronization and small information flow capacity in the brain network !'], ['The complexity of the circuitry of the nervous system of the human brain is still a big challenge to be resolved as it contains about 86 billion neurons and thousands times more synapses .', 'A synapse is a junction between two neurons and it is a mean through which neurons communicate with each other .', 'There are electrical and chemical synapses: An electrical synapse is a physical connection between two neurons which allows electrons to pass through neurons by a very small gap between nerve cells .', 'Electrical synapses are bidirectional and of a local character , happening between neurons whose cells are close .', 'They are believed to contribute to the regulation of synchronization in the brain network .', 'In contrast , chemical synapses are special junctions through which the axon of the pre-synaptic neuron comes close to the post-synaptic cell membrane of another neuron or non-neural cell .', 'In Ref .', '[37] , the authors report on the self-consistent gap junctions and chemical synapses in the connectome of the C . elegans .', 'In our work , we use both kinds of synapses .', 'Following Ref .', '[34] , we endow the nodes ( i . e . neurons for the c . elegans and neural ensembles for the humans ) of the networks with Hindmarsh-Rose brain dynamics [38]:', 'p ˙ = q - a p 3 + b p 2 - n + I ext , q ˙ = c - d p 2 - q , n ˙ = r [ s ( p - p 0 ) - n ] , ( 1 )', 'where p is the membrane potential , q is associated with the fast current , Na+ or K+ , and n with the slow current , for example Ca2+ .', 'The rest of the parameters are defined as a = 1 , b = 3 , c = 1 , d = 5 , s = 4 , p0 = −1 . 6 and Iext = 3 . 25 for which the system exhibits a multi-scale chaotic behavior characterized as spike bursting .', 'r modulates the slow dynamics of the system and was set to 0 . 005 so that each neuron or neural ensemble be chaotic .', 'For these parameters , the HR model enables the spiking-bursting behavior of the membrane potential observed in experiments made with a single neuron in vitro .', 'It is also a relatively simple model that provides a good qualitative description of the many different patterns empirically observed in neural activity .', 'We couple the HR system to create an undirected BDN of Nn neurons connected simultaneously by electrical ( linear diffusive coupling ) and chemical ( nonlinear coupling ) synapses:', 'p ˙ i = q i - a p i 3 + b p i 2 - n i + I ext - g n ( p i - V syn ) ∑ j = 1 N n B i j S ( p j ) - g l ∑ j = 1 N n G i j H ( p j ) , q ˙ i = c - d p i 2 - q i , n ˙ i = r [ s ( p i - p 0 ) - n i ] , ϕ ˙ i = q ˙ i p i - p ˙ i q i p i 2 + q i 2 , i = 1 , … , N n .', '( 2 )', 'In our study , ϕ .', 'i is the instantaneous angular frequency of the i-th neuron and ϕi is the phase defined by the fast variables ( pi , qi ) of the i-th neuron .', 'We consider H ( pi ) = pi and:', 'S ( p j ) = 1 1 + e - λ ( p j - θ syn ) , ( 3 )', 'with θsyn = −0 . 25 , λ = 10 , and Vsyn = 2 to create excitatory BDNs .', 'Eq ( 3 ) is a sigmoidal function that acts as a continuous mechanism for the activation and deactivation of the chemical synapses and , also allows for analytical calculations of the synchronous modes and synchronization manifolds of the coupled system of Eq ( 2 ) [34] .', 'In Eq ( 2 ) , gn is the coupling strength associated to the chemical synapses and gl to the electrical synapses .', 'For the chosen parameters , we have ∣pi∣ < 2 and that ( pi − Vsyn ) is always negative for excitatory networks .', 'If two neurons are connected under an excitatory synapse then , when the presynaptic neuron spikes , it induces the postsynaptic neuron to spike .', 'We adopt only excitatory chemical synapses in our analysis .', 'We use as initial conditions for each neuron i: pi=−1 . 30784489+ηir , qi=−7 . 32183132+ηir , ni=3 . 35299859+ηir and ϕi = 0 , where η i r is a uniformly distributed random number in [0 , 0 . 5] for all i = 1 , … , Nn , following Ref .', '[34] .', 'These initial conditions place the trajectory quickly in the attractor of the dynamics and thus , there is less need to consider longer transients .', 'Gij accounts for the way neurons are electrically ( diffusively ) coupled and it is a Laplacian matrix ( i . e . Gij = Kij − Aij , where A is the binary adjacency matrix of the electrical connections and K is the degree identity matrix based on A ) , and so ∑ j = 1 N n G i j = 0 .', 'By a binary adjacency matrix , we mean an adjacency matrix with entries either 0 ( no connection ) or 1 ( connection ) .', 'Bij is a binary adjacency matrix and describes how the neurons are chemically connected and therefore its diagonal elements are equal to 0 , giving thus ∑ j = 1 N n B i j = k i , where ki is the degree of the i-th neuron , i . e . it represents the number of chemical links that neuron i receives from all other j neurons in the network .', 'A positive ( i . e . 1 ) off-diagonal value in both matrices A , B in row i and column j means that neuron i perturbs neuron j with an intensity given by glGij ( electrical diffusive coupling ) or gnBij ( chemical excitatory coupling ) , respectively .', 'Therefore , the binary adjacency matrices C of the BDNs considered in this work are given by:', 'C = A + B .', 'We numerically integrated Eq ( 2 ) in Fortran 90 using the Euler integration method ( order one ) with time step δt = 0 . 01 .', 'We decided to do so to reduce the numerical complexity and CPU time of the required simulations to feasible levels as a preliminary comparison of trajectories computed for the same parameters ( i . e . δt , initial conditions , etc . ) with integration methods of order 2 , 3 and 4 , revealed similar results .', 'We evolve the dynamics of the brain networks and calculate their two largest Lyapunov exponents λ1 , λ2 for the estimation of the upper bound for MIR , Ic .', 'We use the well-known method of Refs .', '[
['Introduction', 'Results', 'Materials and Methods', 'Discussion']
['We propose a working hypothesis supported by numerical simulations that brain networks evolve based on the principle of the maximization of their internal information flow capacity .', 'We find that synchronous behavior and capacity of information flow of the evolved networks reproduce well the same behaviors observed in the brain dynamical networks of Caenorhabditis elegans and humans , networks of Hindmarsh-Rose neurons with graphs given by these brain networks .', 'We make a strong case to verify our hypothesis by showing that the neural networks with the closest graph distance to the brain networks of Caenorhabditis elegans and humans are the Hindmarsh-Rose neural networks evolved with coupling strengths that maximize information flow capacity .', 'Surprisingly , we find that global neural synchronization levels decrease during brain evolution , reflecting on an underlying global no Hebbian-like evolution process , which is driven by no Hebbian-like learning behaviors for some of the clusters during evolution , and Hebbian-like learning rules for clusters where neurons increase their synchronization .']
['The study of the function of the brain is of primordial importance in neuroscience .', 'Several brain models have been studied so far that take into account higher level functions of the external stimulus and the behavioral response attributed to ensembles of neurons of cortical areas .', 'If the brain learns by maximizing the Mutual Information between stimuli and response , or by updating the internal model of probabilities by using Bayesian techniques , or by minimizing the free-energy , it provides little insight about the dynamical mechanisms appearing in brain networks when evolved based on such principles .', 'In our work we propose a working hypothesis supported by numerical simulations that brain dynamical networks evolve based on the principle of the maximization of their internal information flow capacity , i . e . the upper bound for the information transferred per time unit between any two nodes .', 'We make a strong case to verify our hypothesis by showing that the neural networks with the closest spectral graph distance to the brain networks of Caenorhabditis elegans and humans are the Hindmarsh-Rose neural networks evolved with coupling strengths that maximize information flow capacity .', 'We also find that synchronous behavior and capacity of information flow of the evolved neural networks reproduce well the same behaviors observed in the brain dynamical networks of Caenorhabditis elegans and humans .', 'Finally , we find that global neural synchronization levels decrease during brain network evolution .']
[]