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Mesoscopic model of neuronal system deficits in Multiple Sclerosis
Author’s Accepted Manuscript Mesoscopic Model of Neuronal System Deficits in Multiple Sclerosis Bahareh Safarbali, Fatemeh Hadaeghi, Shahriar Gharibzadeh www.elsevier.com/locate/yjtbi
PII: DOI: Reference:
S0022-5193(16)30196-5 http://dx.doi.org/10.1016/j.jtbi.2016.07.013 YJTBI8738
To appear in: Journal of Theoretical Biology Received date: 9 January 2016 Revised date: 28 June 2016 Accepted date: 7 July 2016 Cite this article as: Bahareh Safarbali, Fatemeh Hadaeghi and Shahriar Gharibzadeh, Mesoscopic Model of Neuronal System Deficits in Multiple S c l e r o s i s , Journal of Theoretical Biology, http://dx.doi.org/10.1016/j.jtbi.2016.07.013 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Mesoscopic Model of Neuronal System Deficits in Multiple Sclerosis Bahareh Safarbalia, Fatemeh Hadaeghia, Shahriar Gharibzadehb a
Complex Systems and Cybernetics Control Laboratory, Biomedical Engineering Faculty, Amirkabir University of
Technology, Tehran, Iran b
Neural and cognitive science Laboratory, Biomedical Engineering Faculty, Amirkabir University of Technology,
Tehran, Iran *
Correspondence: [email protected]
Abstract Multiple Sclerosis (MS) is a devastating autoimmune disease which deteriorates the connections in central nervous system (CNS) through the attacks to oligodendrocytes. Studying its origin and progression, in addition to clinical developments such as MRI brain images, cerebrospinal fluid (CSF) variation and quantitative measures of disability (EDSS), which sought to early diagnosis and efficient therapy, there is an increasing interest in developing computational models using the experimental data obtained from MS patients. From the perspective of mathematical modelling, although the origin of systemic symptoms might be attributed to cellular phenomena in microscopic level such as axonal demyelination, symptoms mainly are observed in macroscopic levels. How to fill the gap between these two levels of system modelling, however, remains as a challenge in systems biology studies. Trying to provide a conceptual framework to bridge between these two levels of modelling in systems biology, we have suggested a mesoscopic model composed of interacting neuronal population, which successfully replicates the changes in neuronal population synchrony due to MS progression. Keywords:
Mesoscopic
Neuronal
Network,
Multiple Sclerosis, Nonlinear dynamics,
Synchronization, Symptom. 1. Introduction
Multiple Sclerosis (MS) is a devastating autoimmune disease which deteriorates the connections in central nervous system (CNS) through the attacks to oligodendrocytes. Oligodendrocytes are 1
known as a kind of glial cells responsible for myelin layers generation around the neurons. Myelin layers, indeed, play a crucial role in facilitation the information transfer between different regions of the brain at normal conditions [1]. However, following an autoimmune attack in MS, the myelin production would be diminished in various regions of the brain. This reduction is known as demyelination and the destroyed collections are referred to demyelination plaque [2]. This devastation mostly reduces the nerve conduction in the central nervous system through which various neurological abnormalities may arise and tend to remit along the time [3]. Demyelination can change the electrophysiological activity of axons and result in abnormal patterns of excitability and electrical activity in different neuronal networks all over the brain. In the normal myelinated neural fibers, a kind of saltatory impulse propagation over the axonal length would be realized due to high densities of sodium channels in Ranvier’s node and rare population of these channels in internodal region [4]. In demyelinated fibers, in contrast, distributed sodium channels provide a path for persistent inward sodium current in internodal regions so that an ectopic firing appeared in neuronal dynamics. Experimentally observed in Waxman studies, sodium channel’s redistribution along the axon alters the pattern of action potential propagation in demyelinated nerves [3]. increases the gain of visual, sensory and motor functions of the nervous system and consequently results in prominent symptoms in MS, such as pain, tingling, double vision, weakness, muscle spasms, etc. [5]. Although more than 2.5 million people around the world currently suffer from MS, the neurobiological basis of the illness is poorly understood and there is no definite treatment proposed for MS [2], [3]. However, in order to find new strategies to early detection of axon demyelination, many electrophysiological tests [6], MRI brain images [7], cerebrospinal fluid (CSF) variation [8], [9] and quantitative measures of disability (EDSS) are investigated in numerous research areas [10], [11]. In addition to take the advantage of diagnostic improvements to study the dynamic aspects of neuronal activity in MS, promotions in the electrical stimulation as a therapeutic strategy has revealed that changes in dynamical patterns of activity in brain neuronal networks might reduce some of MS symptoms such as tremor [12], [13]. Besides the above mentioned clinical developments which sought to early diagnosis and efficient therapies, in recent years, there is an increasing interest in application of nonlinear dynamics in 2
experimental data analysis as well as in developing novel computational models to shed some light on dynamical aspects of MS progression. For the most part, many of the proposed computational models for normal and abnormal brain states in MS are being developed based on microscopic studies of cellular and sub-cellular electrochemical processes. As an example, following the nonlinear analysis of positive and paroxysmal symptoms of illness, Coggan et al, in their innovative study, could provide a mathematical description to model the electrical activity of demyelinated cells. The Proposed model roots in previously designed HodgkinHuxley (H-H) and Morris Lecar (M-L) minimal mathematical representations for neurons. Although using this mathematical description, they could computationally model the effects of demyelination on axonal excitability and axonal conductance, similar to other microscopic models of electrophysiological processes, interactions between neurons and collective behavior of neurons in a connected structure have been ignored in the developed model [14], [5]. Regarding to the fact that the brain is a complex system composed of interactive neuronal ensembles, neuronal demyelination in any of these neuronal networks definitely affects the function of the whole web [15]. Consequently, proposing a mesoscopic model, seems to play a critical role in studying the dynamical behaviors based on neuronal collections and columns [16]. Therefore, in order to develop a computational model at this level, researches on functional and structural connectivity between different regions of the brain seems to be highly productive. There are integrity of findings which reveal changes in both functional and structural connectivity may result in disturbed cognitive functions and motor reactions in several diseases [17]. To be clearer, mesoscopic brain dynamics mention to the dynamics of neuronal populations, networks or columns within cortical regions. It is described by its high complexity, often relating to oscillations at different frequencies and amplitudes, perhaps disturbed by nonlinear behavior. An important feature of mesoscopic modeling is spatio-temporal patterns of activity. Modeling in this level is based on the findings about functional and structure connectivity between different regions of the brain. Changes in this connectivity caused to changes on cognitive and motor functions in brain activity [18], [19]. Computational modeling based on complex system theory has been previously applied to study the change in brain dynamics in case of epilepsy, Alzheimer and some of the mental disorders. According to these studies, normal and abnormal 3
brain dynamics might be associated with the altered patterns of interactions between ensembles of coupled nonlinear dynamic subsystems [20]. The integrity of these researches also confirms that to optimize the rate of information flow between various regions of the brain in the normal state, some of these sub-systems must remain synchronized. In abnormal states such as MS disease, however, either irregular synchronization or desynchronization of activities in different neuronal populations have been reported in resting state (slow wave) studies or during performing cognitive tasks. In mesoscopic level of study, therefore, it has been hypothesized that these unsynchronized patterns have rooted in demyelinated plaque’s formation in various regions of the brain [21]. As a brief, a mesoscopic model of interactive neuronal regions can explain how microscopic change in a number of neurons/neuronal ensembles, leads to macroscopic symptoms such as tremors, fatigue and cognitive problems. In such a multiscale model, the interaction of demyelinated neurons could be implemented as connected abnormal agents in a neuronal network. The impact of abnormal relationship between agents on synchronization of network units could be utilized to study the MS symptoms and its progression. In order to investigate how microscopic changes in a number of neurons/neuronal ensembles may lead the system to show macroscopic symptoms such as tremor, fatigue and cognitive problems, we, developed a minimal mesoscopic model of the brain complex web to study how asynchronous patterns evolve as demyelination is progressing.
2. Model: From the perspective of graph theory, our proposed network is composed of a number of inter related nodes associated with connected neuronal populations in various brain regions so that the structural or functional connections between different populations are represented as links of the graph. To mathematically describe each of neuronal ensembles, we took the advantages of mean field expression of M-L model of neurons which successfully replicates the neuronal dynamics - at the case of demyelination as well as normal condition. M-L model is known as a modified version of the H-H neuron model which not only does include the major ion channels embedded in most 4
axons but also contains detailed information about the voltage-dependent kinetics of the Na and K channels. Similar to pioneer work of H-H, the M-L model has been also proposed based on the electrical conductivity of major ion channels on the membrane of excitable cells and is described by three coupled differential equations. Eq. 1 and Eq. 2 describe this model [14]: (
)
( )( ̅
) ̅
(
) ̅
(
)
(1)
Where v is the mean membrane voltage of a neuronal population. Control parameters such as .
,
,
and
,
show the leak, sodium, potassium and persistent sodium
conductance, respectively. Two other state variables (i.e., w and z) associated with the time-and voltage-dependent activation of potassium and slow sodium channels are also calculated through the following differential equations, Eq. 2 and Eq. 3. ( ) ( )
(2)
( ) ( ) ( ) ( )
(3) (
[ ⁄
(
)]
(4)
)
Where is the rate function and
(5)
and
indicate the threshold and scaling parameters, respectively [14] .
Functional connectivity between regions of the brain is mainly described as a relationship between the neuronal activation patterns arise from different neuronal ensembles [16]. These connections with a prominent role in cognition processes and transmission of information across the brain, can be experimentally extracted from resting state/task fMRI (or multi-channel EEG) records. Taking the advantages of the methods in the time-series correlation studies, dynamic network model of these connections can computationally describe the evolution of functional linked between different brain regions in the resting states [22]. In recent years, imaging techniques such as magnetic resonance imaging (MRI), functional magnetic resonance imaging (fMRI) and diffusion tensor imaging (DTI) have been applied in such brain studies [23]. Although these techniques are mainly focused on investigating the 5
changes in white matter (WM) integrity at the abnormal cases, they provide useful data records to study the network properties of the identified resting or processing nets such as visual, sensorimotor and language networks; where in case of MS disease, one of the strongest domains of evidence relates to abnormality in topological efficiencies (such as small world properties) and synchronous patterns of activity in some of the brain regions [24]. The identified regions of abnormality (listed in table 1) are mainly located on the sub-cortical areas of sensorimotor, visual, default-mode and language.
Table 1- A list of impaired sub-cortical regions of the brain in patients with MS [24].
Systems
Region
SOG.L SOG.R
Visual
CUN.R MOG.L PCL.L PreCG.L
Sensorimotor
PreCG.R PoCG.R PCUN.L PCUN.R
Default-Mode
DCG.R PCG.L IPL.R IFGoperc.L IFGoperc.R
Language
ROL.L IFGtriang.L MFG.L
6
MFG.R
In order to insert the observed functional relations in the proposed network, as stated in Eq. 6, following the generalized coupled map lattices (GCML) paradigm, we considered an interaction term in ML model (Eq. 1) [25], [26]. In this equation, Part Ι indicates the individual dynamic of each neuron (network nodes) and part ΙΙ shows how neuronal ensembles are interacting (network links). ⏟
(
) ̅
( )(
⁄ ⏟
In Eq.6, N is the number of ensembles
∑
)
(
( ̅
) ̅
(
)
)
(6)
and denotes the coupling strength.
Regarding the previously identified sub-cortical regions with high amounts of demyelinated white matter at the instance of MS disease, the proposed not- fully connected network includes 19 nodes and 41 links. Fig. 1 schematically summarizes the overall structure of the network.
Figure 1- Schematic representation of the proposed mesoscopic model. The nodes are those areas of the brain which show significant dysfunction in MS. The links were extracted from experimental functional connectivity matrix reported in [27].
7
2.1. Network synchronization In this study, we extracted two of the network features (i.e., local and global phase synchrony) to quantify the qualitative behavior of functional sub-cortical network in MS in comparison to the control healthy subjects. In the first step, we utilized the synchronization analysis on network as well as dynamical features. Then, in order to quantify the degree of network complexity, we computed the Lyapunov exponent for the synchronization signal arises from the collective behavior of whole network. 2.1.1. Hilbert transform phase synchrony To evaluate the local phase synchrony, the correlation between phases (e.g.,
and
) of the
signals associated with two populations (say x and y) should be computed. It is worth mentioning that the phase synchronization does not depend on the amplitudes of the amplitudes of
and . Clearer, although
and might be statistically independent, the instantaneous phases can be
synchronized. Therefore, we chose to extract the local phase synchronization feature in the Hilbert transformation framework. Hilbert transform is mainly useful to obtain the phase synchronization between different nodes and the deterioration of neurons. This transform is also applicable in chaotic patterns such as brain waves [28]. If we suppose is the observed time series, as stated in Eq.7, ̃( ) shows its Hilbert transform.
( )
( ) |〈
Where
̃( ) ( )
〉|
( )
( ) √〈
( )〉
(7) 〈
( )〉
indicates the calculated phase synchrony which might vary in the interval of [
] so
that 1 stands for completely synchronized time series.
2.1.2. General network synchronization In addition to study the evolution of correlation matrices as disease progresses, we chose to purse the dynamics of global level of network synchrony through a standard deviation measure 8
introduced in [25]. The synchronization of coupled neurons has been considered as a mechanism for connecting spatially distributed features into a coherent object which show an important role in intercommunication between nodes [25]. Following this paradigm, global neuronal synchrony during a simulation is computed by Eq.8 in which the standard deviation temporal synchronization of the system at each time instance,
( ) calculates the
. In our research, this
measurement illustrates the level of synchrony between all nodes (the neuron’s membrane action potential over a long time), and its change shows the effect of demyelination on total performance of the functional network.
(8) ( )
√
[ ∑
( )
{ ∑
(
( )} ]
)
In Eq. 8, N is the number of nodes, and potential. Higher values of
( ) indicates the time series of membrane action
( ) demonstrate the smaller level of global synchronization (i.e.,
plenty of deviation between nodes) while, small ( ) indicates strong synchrony between nodes so that ( )
stands for complete global synchronization.
In order to evaluate the performance of the model, besides the degree of synchrony, a dynamical invariant was investigated to evaluate the network complexity during the diseases progression. Several nonlinear features such as the correlation dimension, fractals dimension and Lyapunov exponent have been commonly reported to quantify the behavioral complexity [29]. In this study, however, we utilized the Lyapunov exponent which has been proven to be the most sensitive feature in dynamical systems to discover the complex higher order behaviors [30]. Lyapunov exponent for a time series (defined in Eq. 9) could be calculated using box counting approach [31]. ( ) ( ) Where
( ) shows the time series and
(9) indicates high complexity in signal and higher
level of regularity in the time series results in and
9
.
To propose a new diagnostic computational index of disease progression, we gradually increased the number of demyelinated ensembles and investigated the effect of its spread on the quantity of Lyapunov exponent extracted from the synchrony signals. 3. Results: In this section, along with a brief review of the role of the number of demyelinated neuronal ensembles on the dynamics of global synchronization in the network, we show how Lyapunov exponent measure of complexity and the network features can provide a computational diagnostic index to grade the MS disease. Initially, we supposed all the neurons are healthy so that the persistent sodium conductance (
) for all neuronal populations were set to one or a value less than 1. To replicate the disease
progression, we then gradually increased the
.
Similar to Coggan original paper, we used the following parameters in all our simulations: C
EL
E Na
EK
m
m
w
w
2
-70
50
-100
w
z
z
z
-1.2 gc
18 gK
-10 g Na
10 gL
0.15
-45
10
0.05
0.1
20
20
2
It is worth mentioning that these values for control conductance parameters were experimentally estimated in [14].
3.1. Effects of increase in persistent sodium conductance: At the instance of demyelination, increase in results in sub-threshold low frequency oscillations so that electrical patterns switch from a phasic spiking to a tonic spiking regime. Fig. 2 illustrates examples of these spontaneous spiking in five destructed nodes (neuronal ensembles) of the network. Regarding the findings in [14], we adjusted the persistent sodium conductance (i.e., ) above a pre-defined threshold to represent the demyelinated axons [14]. Regarding the connection weights, it can be immediately observed that while the ectopic activities emerge in damaged populations, different patterns (mainly abnormal patterns) would be appeared in healthy neuronal populations (see Fig. 2).
10
Figure 2- The action potentials of sample neurons (nodes) of the network (Ni, i=1:8) when N1 and N5 are damaged. The vertical axis shows the membrane voltage in millivolt and the horizontal axis indicates the time in milliseconds. Hyper activity (spiking regime) in N1 and N5 distort the patterns of activity in other interacted neurons to show sub-threshold oscillations mounted on the normal action potentials.
3.2. Effects of potassium blocking: As previously shown in Waxman studies, potassium channels block observed in the demyelinated neurons suppresses the normal action potential [32]. In our model, as depicted in
11
the Fig. 3, potassium block in one of the ensembles affects the other interacted nodes and lead them to show ectopic patterns of activity. It is also consistent with Kapoor research, which experimentally considered the effect of potassium current on genesis of ectopic activity in the demyelinated neurons [33]. Fig. 3 represents the behavior of sample neurons (nodes) of network when N1 is blocked. Moreover, in the other directly connected population (N2) sub-threshold oscillation is appeared but a single action potential response is desired. The other healthy nodes of the network shows such sub- threshold spontaneous activities due to potassium block in a node.
Figure 3- The action potentials of sample neurons (nodes) of the network (N i, i=1:8) when potassium channels are blocked in N1 population. The vertical axis shows the membrane voltage in millivolt and the horizontal
12
axis indicates the time. Hyper activity (spiking regime) in N8, the immediate neighbour of N1 distort the patterns of activity in other interacted neurons to show sub-threshold oscillations mounted on the normal action potentials.
3.3. The effects of increase in persistent sodium channels conductance on phase synchrony: As we noticed in previous section, gradually increasing the number of demyelinated ensembles, we calculated local phase synchrony of the nodes in the network using Hilbert transform of the time series stated in Eq. 7. Fig. 4, illustrate the alterations in local synchrony matrices in that nodes (1), (2), (3) and (4) stand for sub-cortical visual regions and nodes (9), (10), (11), (12) and (13) belong to the sub-cortical default mode areas (see Table.1). A significant abbreviation of phase synchrony of electrical activity in these regions were observed in our simulations. Indeed, since the phase synchrony of these regions is diminishing during disease progression,
is
getting smaller (Fig. 4). These results are consistent with those evidence observed by experimental studies which suggest that demyelination lesions deteriorate the synchronization of different central nervous system (CNS) pathways [27]. The study showed that impaired brain oscillatory activity during a cognitive process is accompanied with such synchrony dysfunctions [27]. In addition, there exist experimental evidence which confirm significant reduction in correlation between activities of some cortical regions in patient with MS [34].
13
Figure 4- Matrices of local phase synchrony in the network during disease progression. (A) All populations are in normal state, (B) 6 ensembles are demyelinated, (C) 12 populations are demyelinated and in (D) all nodes are damaged.
3.4. The effects of increase in persistent sodium channels conductance on global synchronization signal: As demyelination process gradually spreads through the network, we represented the dynamics of the global synchronization signal (Eq. 8) using time series observed from all neuronal populations in the functional network. At normal instance where all populations adjusted to be healthy the complete synchronization of neurons (i.e., ( )
) is expected so that the global
synchronization signal follows a wide action potential like pattern over the time. However, as depicted in Fig. 5, change in electrical pattern of axon response during disease progression significantly alters the quality and quantity of variations in synchronization signals pattern. Our simulation shows that increase in number of demyelinated neuronal ensembles not only does highly diminish the global synchrony (i.e., intensify the amplitude of ( )) but also change the dynamical properties of the global synchronization time series.
14
Figure 5- The global synchronization signals versus time (ms). (A) Complete synchronization when all populations are healthy. (B) Increase in standard deviation and complexity as well as decrease in global synchronization when 10 neuronal ensembles are destroyed. (C) Further diminish and complexity in synchronization of all neurons when all neurons are destroyed.
As we mentioned in 3.1, damaged neurons would also affect the patterns of activities in healthy populations so that the synchronization of undamaged neurons would be highly abbreviated as well (Fig. 6).
15
Figure 6- The synchronization signal of healthy neurons in different states over time. (A) Complete synchronization of all neurons and ( ) when all neurons are healthy. (B) Diminished synchronization in healthy populations when other interacted ensembles are damaged.
3.4.1. The effects of increase in persistent sodium channels conduction on Lyapunov exponent of the global synchronization signal: Fig. 7 shows how the complexity of the global synchronization signals may vary while number of demyelinated populations are increasing. Essentially, in this part, we want to emphasize that the “complexity variation” of synchronization signals was significant in the demyelination development, with any level of destruction. According to the table 2, the amount of Lyapunov exponent is not approaching to the steady state.
16
Figure 7- Changes in largest Lyapunov exponent during the disease progresses.
Table 2- Complexity variations in the demyelination development.
Number of demyelinated neurons
Lyapunov of synchrony signal
0
0.0941
1
0.1012
2
0.1354
3
0.1408
4
0.1482
5
0.1472
6
0.1488
17
7
0.1547
8
0.1525
9
0.1538
10
0.1543
11
0.1569
12
0.1611
13
0.1576
14
0.1645
15
0.1643
16
0.1588
17
0.1605
18
0.1513
19
0.1525
3.5. The effects of increase in persistent sodium channels conductance of one region on general synchronization signal: Different from the previous sub-sections, in this part, we show how change in control parameter associated with the conductivity of persistent sodium channel (i.e.,
) of one ensemble affects
the local and global synchronization of the nodes. In Fig. 8, we chose to gradually enhance the to show similar to increase in number of demyelinated populations, further advances in intensity of demyelination leads to more complex pattern of synchronization signal as well. In addition, it can be obtained that enhance in depth of demyelination in a population have a more significant effect on patterns of activity in healthy ensembles.
18
19
Figure 8- The action potentials of sample neurons (nodes) of the network (N i, i=1:8) when when N1 is damaged and demyelination depth is changing from gNap=1 to gNap=3 (a: gNap=1, b: gNap=1.5, c: gNap=2, d: gNap=3). The last plot in each section shows the global synchronization signals of all populations over time.
4. Discussion We proposed a mesoscopic network based on the electrophysiologically plausible model of neurons and functional connectivity in brain networks to investigate the symptoms in multiple sclerosis. Although the origin of systemic symptoms might be attributed to cellular phenomena in microscopic level such as axonal demyelination, their representations are in macroscopic levels and there is a gap between these two levels from the modeling perspective. Such a mesoscopic model, therefore, can explain how microscopic change in a number of neurons lead to macroscopic symptoms such as tremors, fatigue and cognitive problems. Mesoscopic models are mainly developed based on the functional connections and neuronal interactions in the brain networks. In such models, the interaction of demyelinated neurons can be implemented as connected abnormal agents in a neuronal network. Thereafter, the impact of abnormal relationship between agents on synchronization of network units can be applied to study the higher level MS symptoms. Biological systems are structured at scales of many orders of degree in space and time. To link a cell as a microscopic level to a biological function of an organ or a disease as a macroscopic pattern, it is crucial to understand how the cells interact with each other to result in a function. Based on the experimental evidence, a model can be established either using the top-down approach or the bottom-up approach. In bottom up method in system biology, the mesoscopic level feature is the result of cells interaction as the microscopic elements. In fact, extracting the main feature of large network as a result of variation in node’s dynamics, is a kind of connection between two microscopic and mesoscopic levels. There is not any direct way to go from one scale to another, but the proposed model can provide the useful methods that may be used to bridge the gaps [35]. In the interim, we investigated the neuronal network for MS using graph theoretical approaches and Morris Lecar model of excitable cells. The proposed network is composed of a number of nodes, which represents the connected neuronal populations in various areas of the brain. The functional connections between different regions are then represented as links between these
20
nodes. In our model, increase in the persistent sodium conductance plays a determining role in demyelination progression. Our model represents that, in the demyelinated conditions, any increase in persistent sodium conductance results in a kind of sub-threshold low frequency oscillations, which might change axon’s electrochemical behaviors. Due to these oscillations, electrical patterns shifts from phasic spiking to a tonic spiking regime or lead the system to exhibit spontaneous spikes.
Neurophysiological and neuroimaging techniques confirm that the abnormal patterns of synchronization in different central nervous system (CNS) pathways is mainly caused by demyelinating lesions which may lead to either the slowing down the brain oscillatory activity or decreasing the efficiency of the cognitive processing. Therefore, impairment in cognitive processing in patients with MS may be reflected in abbreviation in synchronization observed in the proposed mesoscopic model. For instance, low level of synchronization in binocular movements in patients with internuclear ophthalmoparesis results in position discrepancy between the eyes and consequently double vision problems. In this regard, our simulation shows that local and global impaired synchronization patterns not only does bridge the gap between medium and high levels of activity (i.e., MS symptoms) but also provides a computational diagnostic index to grade the disease progression. Most of previously proposed computational models have been developed based on microscopic studies of cellular and sub-cellular electrochemical processes. Microscopic models of electrophysiological processes in the cell’s deal with the constraint of observing a single neuron in an isolated state so that the interactions between neurons and collective behavior of neurons in the connected structure are ignored. For instance, previously, Coggan, et al, developed a computational model of a demyelinated axon in which change in the ratio of channel conductivities (e.g.,
⁄
) can lead the system to transit between distinct axonal spiking
patterns (e.g., failure, single-spike, AD, and spontaneous) [5]. However, correlation of the
21
activities in microscopic level with the symptoms observed at higher level seems to be beyond the scope of the mentioned study. The proposed model of interactive neuronal regions, therefore, is a novel representation of a network which can help to fill the gap between macroscopic symptoms and microscopic changes in electrophysiological properties of neurons. To improve our model in future works, it might be helpful to consider the channel’s conductance as time variables in a dynamic network formalism. Since the symptoms vary in different situations, this developed network can be replaced by a proper model based on the situation. Also, it would be quite feasible to think to import the propagation dynamics associated with the demyelinated neurons in the model and provide a spatio-temporal framework to study the novel patterns of activity in MS.
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Strogatz, S.H., Exploring complex networks. Nature, 2001. 410(6825): p. 268-276. Schoonheim, M.M., et al., Gender-related differences in functional connectivity in multiple sclerosis. Multiple sclerosis journal, 2012. 18(2): p. 164-173. Freeman, W., Neurodynamics: an exploration in mesoscopic brain dynamics. 2012: Springer Science & Business Media. Haken, H., Mesoscopic levels in science-some comments. Micro-Meso-Macro: Addressing Complex Systems Couplings, 2005: p. 19-24. Liljenström, H. and U. Svedin, Micro, meso, macro: addressing complex systems couplings. 2005: World Scientific. Bullmore, E. and O. Sporns, Complex brain networks: graph theoretical analysis of structural and functional systems. Nature Reviews Neuroscience, 2009. 10(3): p. 186-198. Uhlhaas, P.J. and W. Singer, Neural synchrony in brain disorders: relevance for cognitive dysfunctions and pathophysiology. Neuron, 2006. 52(1): p. 155-168. Achard, S., et al., A resilient, low-frequency, small-world human brain functional network with highly connected association cortical hubs. The Journal of neuroscience, 2006. 26(1): p. 63-72. Genova, H.M., et al., Examination of cognitive fatigue in multiple sclerosis using functional magnetic resonance imaging and diffusion tensor imaging. 2013. Shu, N., et al., Diffusion tensor tractography reveals disrupted topological efficiency in white matter structural networks in multiple sclerosis. Cerebral Cortex, 2011. 21(11): p. 2565-2577. Mao-Sheng, W., H. Zhong-Huai, and X. Hou-Wen, Synchronization and coherence resonance in chaotic neural networks. Chinese Physics, 2006. 15(11): p. 2553. Qing-Yun, W., L. Qi-Shao, and W. Hai-Xia, Transition to complete synchronization via nearsynchronization in two coupled chaotic neurons. Chinese Physics, 2005. 14(11): p. 2189. Arrondo, G., et al., Abnormalities in brain synchronization are correlated with cognitive impairment in multiple sclerosis. Multiple Sclerosis, 2009. Quiroga, R.Q., et al., Performance of different synchronization measures in real data: a case study on electroencephalographic signals. Physical Review E, 2002. 65(4): p. 041903. Porcher, R. and G. Thomas, Estimating Lyapunov exponents in biomedical time series. Physical Review E, 2001. 64(1): p. 010902. Rosenstein, M.T., J.J. Collins, and C.J. De Luca, A practical method for calculating largest Lyapunov exponents from small data sets. Physica D: Nonlinear Phenomena, 1993. 65(1): p. 117134. Jiménez, A., C. Vara de Rey, and A. Torres, Effect of parameter calculation in direct estimation of the Lyapunov exponent in short time series. Discrete Dynamics in Nature and Society, 2002. 7(1): p. 41-52. Waxman, S.G., Demyelination in spinal cord injury. Journal of the neurological sciences, 1989. 91(1): p. 1-14. Kapoor, R., et al., Internodal potassium currents can generate ectopic impulses in mammalian myelinated axons. Brain research, 1993. 611(1): p. 165-169. Zhou, Y., et al., Functional homotopic changes in multiple sclerosis with resting-state functional MR imaging. American Journal of Neuroradiology, 2013. 34(6): p. 1180-1187. Qu, Z., et al., Multi-scale modeling in biology: how to bridge the gaps between scales? Progress in biophysics and molecular biology, 2011. 107(1): p. 21-31. Tewarie, P., et al., Cognitive and clinical dysfunction, altered MEG resting-state networks and thalamic atrophy in multiple sclerosis. 2013. Frohman, T.C., et al., Symptomatic therapy in multiple sclerosis. Therapeutic advances in neurological disorders, 2011: p. 1756285611400658. Bjartmar, C., X. Yin, and B.D. Trapp, Axonal pathology in myelin disorders. Journal of neurocytology, 1999. 28(4-5): p. 383-395. Love, S., Demyelinating diseases. Journal of clinical pathology, 2006. 59(11): p. 1151-1159. 23
Highlights
A mesoscopic model composed of interacting neuronal population has been proposed for Multiple Sclerosis (MS) progression.
The model successfully replicate the changes in neuronal population synchrony due to MS progression.
The proposed model provides a computational framework to fill the gap between cellular and behavioral levels of system modeling.
The correlation between changes in general synchronization signal and the MS symptoms are successfully represent in the model.
The proposed model could also represent the change in complexity of the interactions between neuronal population in MS.
24
PII: DOI: Reference:
S0022-5193(16)30196-5 http://dx.doi.org/10.1016/j.jtbi.2016.07.013 YJTBI8738
To appear in: Journal of Theoretical Biology Received date: 9 January 2016 Revised date: 28 June 2016 Accepted date: 7 July 2016 Cite this article as: Bahareh Safarbali, Fatemeh Hadaeghi and Shahriar Gharibzadeh, Mesoscopic Model of Neuronal System Deficits in Multiple S c l e r o s i s , Journal of Theoretical Biology, http://dx.doi.org/10.1016/j.jtbi.2016.07.013 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Mesoscopic Model of Neuronal System Deficits in Multiple Sclerosis Bahareh Safarbalia, Fatemeh Hadaeghia, Shahriar Gharibzadehb a
Complex Systems and Cybernetics Control Laboratory, Biomedical Engineering Faculty, Amirkabir University of
Technology, Tehran, Iran b
Neural and cognitive science Laboratory, Biomedical Engineering Faculty, Amirkabir University of Technology,
Tehran, Iran *
Correspondence: [email protected]
Abstract Multiple Sclerosis (MS) is a devastating autoimmune disease which deteriorates the connections in central nervous system (CNS) through the attacks to oligodendrocytes. Studying its origin and progression, in addition to clinical developments such as MRI brain images, cerebrospinal fluid (CSF) variation and quantitative measures of disability (EDSS), which sought to early diagnosis and efficient therapy, there is an increasing interest in developing computational models using the experimental data obtained from MS patients. From the perspective of mathematical modelling, although the origin of systemic symptoms might be attributed to cellular phenomena in microscopic level such as axonal demyelination, symptoms mainly are observed in macroscopic levels. How to fill the gap between these two levels of system modelling, however, remains as a challenge in systems biology studies. Trying to provide a conceptual framework to bridge between these two levels of modelling in systems biology, we have suggested a mesoscopic model composed of interacting neuronal population, which successfully replicates the changes in neuronal population synchrony due to MS progression. Keywords:
Mesoscopic
Neuronal
Network,
Multiple Sclerosis, Nonlinear dynamics,
Synchronization, Symptom. 1. Introduction
Multiple Sclerosis (MS) is a devastating autoimmune disease which deteriorates the connections in central nervous system (CNS) through the attacks to oligodendrocytes. Oligodendrocytes are 1
known as a kind of glial cells responsible for myelin layers generation around the neurons. Myelin layers, indeed, play a crucial role in facilitation the information transfer between different regions of the brain at normal conditions [1]. However, following an autoimmune attack in MS, the myelin production would be diminished in various regions of the brain. This reduction is known as demyelination and the destroyed collections are referred to demyelination plaque [2]. This devastation mostly reduces the nerve conduction in the central nervous system through which various neurological abnormalities may arise and tend to remit along the time [3]. Demyelination can change the electrophysiological activity of axons and result in abnormal patterns of excitability and electrical activity in different neuronal networks all over the brain. In the normal myelinated neural fibers, a kind of saltatory impulse propagation over the axonal length would be realized due to high densities of sodium channels in Ranvier’s node and rare population of these channels in internodal region [4]. In demyelinated fibers, in contrast, distributed sodium channels provide a path for persistent inward sodium current in internodal regions so that an ectopic firing appeared in neuronal dynamics. Experimentally observed in Waxman studies, sodium channel’s redistribution along the axon alters the pattern of action potential propagation in demyelinated nerves [3]. increases the gain of visual, sensory and motor functions of the nervous system and consequently results in prominent symptoms in MS, such as pain, tingling, double vision, weakness, muscle spasms, etc. [5]. Although more than 2.5 million people around the world currently suffer from MS, the neurobiological basis of the illness is poorly understood and there is no definite treatment proposed for MS [2], [3]. However, in order to find new strategies to early detection of axon demyelination, many electrophysiological tests [6], MRI brain images [7], cerebrospinal fluid (CSF) variation [8], [9] and quantitative measures of disability (EDSS) are investigated in numerous research areas [10], [11]. In addition to take the advantage of diagnostic improvements to study the dynamic aspects of neuronal activity in MS, promotions in the electrical stimulation as a therapeutic strategy has revealed that changes in dynamical patterns of activity in brain neuronal networks might reduce some of MS symptoms such as tremor [12], [13]. Besides the above mentioned clinical developments which sought to early diagnosis and efficient therapies, in recent years, there is an increasing interest in application of nonlinear dynamics in 2
experimental data analysis as well as in developing novel computational models to shed some light on dynamical aspects of MS progression. For the most part, many of the proposed computational models for normal and abnormal brain states in MS are being developed based on microscopic studies of cellular and sub-cellular electrochemical processes. As an example, following the nonlinear analysis of positive and paroxysmal symptoms of illness, Coggan et al, in their innovative study, could provide a mathematical description to model the electrical activity of demyelinated cells. The Proposed model roots in previously designed HodgkinHuxley (H-H) and Morris Lecar (M-L) minimal mathematical representations for neurons. Although using this mathematical description, they could computationally model the effects of demyelination on axonal excitability and axonal conductance, similar to other microscopic models of electrophysiological processes, interactions between neurons and collective behavior of neurons in a connected structure have been ignored in the developed model [14], [5]. Regarding to the fact that the brain is a complex system composed of interactive neuronal ensembles, neuronal demyelination in any of these neuronal networks definitely affects the function of the whole web [15]. Consequently, proposing a mesoscopic model, seems to play a critical role in studying the dynamical behaviors based on neuronal collections and columns [16]. Therefore, in order to develop a computational model at this level, researches on functional and structural connectivity between different regions of the brain seems to be highly productive. There are integrity of findings which reveal changes in both functional and structural connectivity may result in disturbed cognitive functions and motor reactions in several diseases [17]. To be clearer, mesoscopic brain dynamics mention to the dynamics of neuronal populations, networks or columns within cortical regions. It is described by its high complexity, often relating to oscillations at different frequencies and amplitudes, perhaps disturbed by nonlinear behavior. An important feature of mesoscopic modeling is spatio-temporal patterns of activity. Modeling in this level is based on the findings about functional and structure connectivity between different regions of the brain. Changes in this connectivity caused to changes on cognitive and motor functions in brain activity [18], [19]. Computational modeling based on complex system theory has been previously applied to study the change in brain dynamics in case of epilepsy, Alzheimer and some of the mental disorders. According to these studies, normal and abnormal 3
brain dynamics might be associated with the altered patterns of interactions between ensembles of coupled nonlinear dynamic subsystems [20]. The integrity of these researches also confirms that to optimize the rate of information flow between various regions of the brain in the normal state, some of these sub-systems must remain synchronized. In abnormal states such as MS disease, however, either irregular synchronization or desynchronization of activities in different neuronal populations have been reported in resting state (slow wave) studies or during performing cognitive tasks. In mesoscopic level of study, therefore, it has been hypothesized that these unsynchronized patterns have rooted in demyelinated plaque’s formation in various regions of the brain [21]. As a brief, a mesoscopic model of interactive neuronal regions can explain how microscopic change in a number of neurons/neuronal ensembles, leads to macroscopic symptoms such as tremors, fatigue and cognitive problems. In such a multiscale model, the interaction of demyelinated neurons could be implemented as connected abnormal agents in a neuronal network. The impact of abnormal relationship between agents on synchronization of network units could be utilized to study the MS symptoms and its progression. In order to investigate how microscopic changes in a number of neurons/neuronal ensembles may lead the system to show macroscopic symptoms such as tremor, fatigue and cognitive problems, we, developed a minimal mesoscopic model of the brain complex web to study how asynchronous patterns evolve as demyelination is progressing.
2. Model: From the perspective of graph theory, our proposed network is composed of a number of inter related nodes associated with connected neuronal populations in various brain regions so that the structural or functional connections between different populations are represented as links of the graph. To mathematically describe each of neuronal ensembles, we took the advantages of mean field expression of M-L model of neurons which successfully replicates the neuronal dynamics - at the case of demyelination as well as normal condition. M-L model is known as a modified version of the H-H neuron model which not only does include the major ion channels embedded in most 4
axons but also contains detailed information about the voltage-dependent kinetics of the Na and K channels. Similar to pioneer work of H-H, the M-L model has been also proposed based on the electrical conductivity of major ion channels on the membrane of excitable cells and is described by three coupled differential equations. Eq. 1 and Eq. 2 describe this model [14]: (
)
( )( ̅
) ̅
(
) ̅
(
)
(1)
Where v is the mean membrane voltage of a neuronal population. Control parameters such as .
,
,
and
,
show the leak, sodium, potassium and persistent sodium
conductance, respectively. Two other state variables (i.e., w and z) associated with the time-and voltage-dependent activation of potassium and slow sodium channels are also calculated through the following differential equations, Eq. 2 and Eq. 3. ( ) ( )
(2)
( ) ( ) ( ) ( )
(3) (
[ ⁄
(
)]
(4)
)
Where is the rate function and
(5)
and
indicate the threshold and scaling parameters, respectively [14] .
Functional connectivity between regions of the brain is mainly described as a relationship between the neuronal activation patterns arise from different neuronal ensembles [16]. These connections with a prominent role in cognition processes and transmission of information across the brain, can be experimentally extracted from resting state/task fMRI (or multi-channel EEG) records. Taking the advantages of the methods in the time-series correlation studies, dynamic network model of these connections can computationally describe the evolution of functional linked between different brain regions in the resting states [22]. In recent years, imaging techniques such as magnetic resonance imaging (MRI), functional magnetic resonance imaging (fMRI) and diffusion tensor imaging (DTI) have been applied in such brain studies [23]. Although these techniques are mainly focused on investigating the 5
changes in white matter (WM) integrity at the abnormal cases, they provide useful data records to study the network properties of the identified resting or processing nets such as visual, sensorimotor and language networks; where in case of MS disease, one of the strongest domains of evidence relates to abnormality in topological efficiencies (such as small world properties) and synchronous patterns of activity in some of the brain regions [24]. The identified regions of abnormality (listed in table 1) are mainly located on the sub-cortical areas of sensorimotor, visual, default-mode and language.
Table 1- A list of impaired sub-cortical regions of the brain in patients with MS [24].
Systems
Region
SOG.L SOG.R
Visual
CUN.R MOG.L PCL.L PreCG.L
Sensorimotor
PreCG.R PoCG.R PCUN.L PCUN.R
Default-Mode
DCG.R PCG.L IPL.R IFGoperc.L IFGoperc.R
Language
ROL.L IFGtriang.L MFG.L
6
MFG.R
In order to insert the observed functional relations in the proposed network, as stated in Eq. 6, following the generalized coupled map lattices (GCML) paradigm, we considered an interaction term in ML model (Eq. 1) [25], [26]. In this equation, Part Ι indicates the individual dynamic of each neuron (network nodes) and part ΙΙ shows how neuronal ensembles are interacting (network links). ⏟
(
) ̅
( )(
⁄ ⏟
In Eq.6, N is the number of ensembles
∑
)
(
( ̅
) ̅
(
)
)
(6)
and denotes the coupling strength.
Regarding the previously identified sub-cortical regions with high amounts of demyelinated white matter at the instance of MS disease, the proposed not- fully connected network includes 19 nodes and 41 links. Fig. 1 schematically summarizes the overall structure of the network.
Figure 1- Schematic representation of the proposed mesoscopic model. The nodes are those areas of the brain which show significant dysfunction in MS. The links were extracted from experimental functional connectivity matrix reported in [27].
7
2.1. Network synchronization In this study, we extracted two of the network features (i.e., local and global phase synchrony) to quantify the qualitative behavior of functional sub-cortical network in MS in comparison to the control healthy subjects. In the first step, we utilized the synchronization analysis on network as well as dynamical features. Then, in order to quantify the degree of network complexity, we computed the Lyapunov exponent for the synchronization signal arises from the collective behavior of whole network. 2.1.1. Hilbert transform phase synchrony To evaluate the local phase synchrony, the correlation between phases (e.g.,
and
) of the
signals associated with two populations (say x and y) should be computed. It is worth mentioning that the phase synchronization does not depend on the amplitudes of the amplitudes of
and . Clearer, although
and might be statistically independent, the instantaneous phases can be
synchronized. Therefore, we chose to extract the local phase synchronization feature in the Hilbert transformation framework. Hilbert transform is mainly useful to obtain the phase synchronization between different nodes and the deterioration of neurons. This transform is also applicable in chaotic patterns such as brain waves [28]. If we suppose is the observed time series, as stated in Eq.7, ̃( ) shows its Hilbert transform.
( )
( ) |〈
Where
̃( ) ( )
〉|
( )
( ) √〈
( )〉
(7) 〈
( )〉
indicates the calculated phase synchrony which might vary in the interval of [
] so
that 1 stands for completely synchronized time series.
2.1.2. General network synchronization In addition to study the evolution of correlation matrices as disease progresses, we chose to purse the dynamics of global level of network synchrony through a standard deviation measure 8
introduced in [25]. The synchronization of coupled neurons has been considered as a mechanism for connecting spatially distributed features into a coherent object which show an important role in intercommunication between nodes [25]. Following this paradigm, global neuronal synchrony during a simulation is computed by Eq.8 in which the standard deviation temporal synchronization of the system at each time instance,
( ) calculates the
. In our research, this
measurement illustrates the level of synchrony between all nodes (the neuron’s membrane action potential over a long time), and its change shows the effect of demyelination on total performance of the functional network.
(8) ( )
√
[ ∑
( )
{ ∑
(
( )} ]
)
In Eq. 8, N is the number of nodes, and potential. Higher values of
( ) indicates the time series of membrane action
( ) demonstrate the smaller level of global synchronization (i.e.,
plenty of deviation between nodes) while, small ( ) indicates strong synchrony between nodes so that ( )
stands for complete global synchronization.
In order to evaluate the performance of the model, besides the degree of synchrony, a dynamical invariant was investigated to evaluate the network complexity during the diseases progression. Several nonlinear features such as the correlation dimension, fractals dimension and Lyapunov exponent have been commonly reported to quantify the behavioral complexity [29]. In this study, however, we utilized the Lyapunov exponent which has been proven to be the most sensitive feature in dynamical systems to discover the complex higher order behaviors [30]. Lyapunov exponent for a time series (defined in Eq. 9) could be calculated using box counting approach [31]. ( ) ( ) Where
( ) shows the time series and
(9) indicates high complexity in signal and higher
level of regularity in the time series results in and
9
.
To propose a new diagnostic computational index of disease progression, we gradually increased the number of demyelinated ensembles and investigated the effect of its spread on the quantity of Lyapunov exponent extracted from the synchrony signals. 3. Results: In this section, along with a brief review of the role of the number of demyelinated neuronal ensembles on the dynamics of global synchronization in the network, we show how Lyapunov exponent measure of complexity and the network features can provide a computational diagnostic index to grade the MS disease. Initially, we supposed all the neurons are healthy so that the persistent sodium conductance (
) for all neuronal populations were set to one or a value less than 1. To replicate the disease
progression, we then gradually increased the
.
Similar to Coggan original paper, we used the following parameters in all our simulations: C
EL
E Na
EK
m
m
w
w
2
-70
50
-100
w
z
z
z
-1.2 gc
18 gK
-10 g Na
10 gL
0.15
-45
10
0.05
0.1
20
20
2
It is worth mentioning that these values for control conductance parameters were experimentally estimated in [14].
3.1. Effects of increase in persistent sodium conductance: At the instance of demyelination, increase in results in sub-threshold low frequency oscillations so that electrical patterns switch from a phasic spiking to a tonic spiking regime. Fig. 2 illustrates examples of these spontaneous spiking in five destructed nodes (neuronal ensembles) of the network. Regarding the findings in [14], we adjusted the persistent sodium conductance (i.e., ) above a pre-defined threshold to represent the demyelinated axons [14]. Regarding the connection weights, it can be immediately observed that while the ectopic activities emerge in damaged populations, different patterns (mainly abnormal patterns) would be appeared in healthy neuronal populations (see Fig. 2).
10
Figure 2- The action potentials of sample neurons (nodes) of the network (Ni, i=1:8) when N1 and N5 are damaged. The vertical axis shows the membrane voltage in millivolt and the horizontal axis indicates the time in milliseconds. Hyper activity (spiking regime) in N1 and N5 distort the patterns of activity in other interacted neurons to show sub-threshold oscillations mounted on the normal action potentials.
3.2. Effects of potassium blocking: As previously shown in Waxman studies, potassium channels block observed in the demyelinated neurons suppresses the normal action potential [32]. In our model, as depicted in
11
the Fig. 3, potassium block in one of the ensembles affects the other interacted nodes and lead them to show ectopic patterns of activity. It is also consistent with Kapoor research, which experimentally considered the effect of potassium current on genesis of ectopic activity in the demyelinated neurons [33]. Fig. 3 represents the behavior of sample neurons (nodes) of network when N1 is blocked. Moreover, in the other directly connected population (N2) sub-threshold oscillation is appeared but a single action potential response is desired. The other healthy nodes of the network shows such sub- threshold spontaneous activities due to potassium block in a node.
Figure 3- The action potentials of sample neurons (nodes) of the network (N i, i=1:8) when potassium channels are blocked in N1 population. The vertical axis shows the membrane voltage in millivolt and the horizontal
12
axis indicates the time. Hyper activity (spiking regime) in N8, the immediate neighbour of N1 distort the patterns of activity in other interacted neurons to show sub-threshold oscillations mounted on the normal action potentials.
3.3. The effects of increase in persistent sodium channels conductance on phase synchrony: As we noticed in previous section, gradually increasing the number of demyelinated ensembles, we calculated local phase synchrony of the nodes in the network using Hilbert transform of the time series stated in Eq. 7. Fig. 4, illustrate the alterations in local synchrony matrices in that nodes (1), (2), (3) and (4) stand for sub-cortical visual regions and nodes (9), (10), (11), (12) and (13) belong to the sub-cortical default mode areas (see Table.1). A significant abbreviation of phase synchrony of electrical activity in these regions were observed in our simulations. Indeed, since the phase synchrony of these regions is diminishing during disease progression,
is
getting smaller (Fig. 4). These results are consistent with those evidence observed by experimental studies which suggest that demyelination lesions deteriorate the synchronization of different central nervous system (CNS) pathways [27]. The study showed that impaired brain oscillatory activity during a cognitive process is accompanied with such synchrony dysfunctions [27]. In addition, there exist experimental evidence which confirm significant reduction in correlation between activities of some cortical regions in patient with MS [34].
13
Figure 4- Matrices of local phase synchrony in the network during disease progression. (A) All populations are in normal state, (B) 6 ensembles are demyelinated, (C) 12 populations are demyelinated and in (D) all nodes are damaged.
3.4. The effects of increase in persistent sodium channels conductance on global synchronization signal: As demyelination process gradually spreads through the network, we represented the dynamics of the global synchronization signal (Eq. 8) using time series observed from all neuronal populations in the functional network. At normal instance where all populations adjusted to be healthy the complete synchronization of neurons (i.e., ( )
) is expected so that the global
synchronization signal follows a wide action potential like pattern over the time. However, as depicted in Fig. 5, change in electrical pattern of axon response during disease progression significantly alters the quality and quantity of variations in synchronization signals pattern. Our simulation shows that increase in number of demyelinated neuronal ensembles not only does highly diminish the global synchrony (i.e., intensify the amplitude of ( )) but also change the dynamical properties of the global synchronization time series.
14
Figure 5- The global synchronization signals versus time (ms). (A) Complete synchronization when all populations are healthy. (B) Increase in standard deviation and complexity as well as decrease in global synchronization when 10 neuronal ensembles are destroyed. (C) Further diminish and complexity in synchronization of all neurons when all neurons are destroyed.
As we mentioned in 3.1, damaged neurons would also affect the patterns of activities in healthy populations so that the synchronization of undamaged neurons would be highly abbreviated as well (Fig. 6).
15
Figure 6- The synchronization signal of healthy neurons in different states over time. (A) Complete synchronization of all neurons and ( ) when all neurons are healthy. (B) Diminished synchronization in healthy populations when other interacted ensembles are damaged.
3.4.1. The effects of increase in persistent sodium channels conduction on Lyapunov exponent of the global synchronization signal: Fig. 7 shows how the complexity of the global synchronization signals may vary while number of demyelinated populations are increasing. Essentially, in this part, we want to emphasize that the “complexity variation” of synchronization signals was significant in the demyelination development, with any level of destruction. According to the table 2, the amount of Lyapunov exponent is not approaching to the steady state.
16
Figure 7- Changes in largest Lyapunov exponent during the disease progresses.
Table 2- Complexity variations in the demyelination development.
Number of demyelinated neurons
Lyapunov of synchrony signal
0
0.0941
1
0.1012
2
0.1354
3
0.1408
4
0.1482
5
0.1472
6
0.1488
17
7
0.1547
8
0.1525
9
0.1538
10
0.1543
11
0.1569
12
0.1611
13
0.1576
14
0.1645
15
0.1643
16
0.1588
17
0.1605
18
0.1513
19
0.1525
3.5. The effects of increase in persistent sodium channels conductance of one region on general synchronization signal: Different from the previous sub-sections, in this part, we show how change in control parameter associated with the conductivity of persistent sodium channel (i.e.,
) of one ensemble affects
the local and global synchronization of the nodes. In Fig. 8, we chose to gradually enhance the to show similar to increase in number of demyelinated populations, further advances in intensity of demyelination leads to more complex pattern of synchronization signal as well. In addition, it can be obtained that enhance in depth of demyelination in a population have a more significant effect on patterns of activity in healthy ensembles.
18
19
Figure 8- The action potentials of sample neurons (nodes) of the network (N i, i=1:8) when when N1 is damaged and demyelination depth is changing from gNap=1 to gNap=3 (a: gNap=1, b: gNap=1.5, c: gNap=2, d: gNap=3). The last plot in each section shows the global synchronization signals of all populations over time.
4. Discussion We proposed a mesoscopic network based on the electrophysiologically plausible model of neurons and functional connectivity in brain networks to investigate the symptoms in multiple sclerosis. Although the origin of systemic symptoms might be attributed to cellular phenomena in microscopic level such as axonal demyelination, their representations are in macroscopic levels and there is a gap between these two levels from the modeling perspective. Such a mesoscopic model, therefore, can explain how microscopic change in a number of neurons lead to macroscopic symptoms such as tremors, fatigue and cognitive problems. Mesoscopic models are mainly developed based on the functional connections and neuronal interactions in the brain networks. In such models, the interaction of demyelinated neurons can be implemented as connected abnormal agents in a neuronal network. Thereafter, the impact of abnormal relationship between agents on synchronization of network units can be applied to study the higher level MS symptoms. Biological systems are structured at scales of many orders of degree in space and time. To link a cell as a microscopic level to a biological function of an organ or a disease as a macroscopic pattern, it is crucial to understand how the cells interact with each other to result in a function. Based on the experimental evidence, a model can be established either using the top-down approach or the bottom-up approach. In bottom up method in system biology, the mesoscopic level feature is the result of cells interaction as the microscopic elements. In fact, extracting the main feature of large network as a result of variation in node’s dynamics, is a kind of connection between two microscopic and mesoscopic levels. There is not any direct way to go from one scale to another, but the proposed model can provide the useful methods that may be used to bridge the gaps [35]. In the interim, we investigated the neuronal network for MS using graph theoretical approaches and Morris Lecar model of excitable cells. The proposed network is composed of a number of nodes, which represents the connected neuronal populations in various areas of the brain. The functional connections between different regions are then represented as links between these
20
nodes. In our model, increase in the persistent sodium conductance plays a determining role in demyelination progression. Our model represents that, in the demyelinated conditions, any increase in persistent sodium conductance results in a kind of sub-threshold low frequency oscillations, which might change axon’s electrochemical behaviors. Due to these oscillations, electrical patterns shifts from phasic spiking to a tonic spiking regime or lead the system to exhibit spontaneous spikes.
Neurophysiological and neuroimaging techniques confirm that the abnormal patterns of synchronization in different central nervous system (CNS) pathways is mainly caused by demyelinating lesions which may lead to either the slowing down the brain oscillatory activity or decreasing the efficiency of the cognitive processing. Therefore, impairment in cognitive processing in patients with MS may be reflected in abbreviation in synchronization observed in the proposed mesoscopic model. For instance, low level of synchronization in binocular movements in patients with internuclear ophthalmoparesis results in position discrepancy between the eyes and consequently double vision problems. In this regard, our simulation shows that local and global impaired synchronization patterns not only does bridge the gap between medium and high levels of activity (i.e., MS symptoms) but also provides a computational diagnostic index to grade the disease progression. Most of previously proposed computational models have been developed based on microscopic studies of cellular and sub-cellular electrochemical processes. Microscopic models of electrophysiological processes in the cell’s deal with the constraint of observing a single neuron in an isolated state so that the interactions between neurons and collective behavior of neurons in the connected structure are ignored. For instance, previously, Coggan, et al, developed a computational model of a demyelinated axon in which change in the ratio of channel conductivities (e.g.,
⁄
) can lead the system to transit between distinct axonal spiking
patterns (e.g., failure, single-spike, AD, and spontaneous) [5]. However, correlation of the
21
activities in microscopic level with the symptoms observed at higher level seems to be beyond the scope of the mentioned study. The proposed model of interactive neuronal regions, therefore, is a novel representation of a network which can help to fill the gap between macroscopic symptoms and microscopic changes in electrophysiological properties of neurons. To improve our model in future works, it might be helpful to consider the channel’s conductance as time variables in a dynamic network formalism. Since the symptoms vary in different situations, this developed network can be replaced by a proper model based on the situation. Also, it would be quite feasible to think to import the propagation dynamics associated with the demyelinated neurons in the model and provide a spatio-temporal framework to study the novel patterns of activity in MS.
5. References 1. 2. 3. 4. 5.
6. 7.
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Highlights
A mesoscopic model composed of interacting neuronal population has been proposed for Multiple Sclerosis (MS) progression.
The model successfully replicate the changes in neuronal population synchrony due to MS progression.
The proposed model provides a computational framework to fill the gap between cellular and behavioral levels of system modeling.
The correlation between changes in general synchronization signal and the MS symptoms are successfully represent in the model.
The proposed model could also represent the change in complexity of the interactions between neuronal population in MS.
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