Structural, functional and dynamical properties of a lognormal network of bursting neurons

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(1)UNIVERSIDADE DE SÃO PAULO INSTITUTO DE FÍSICA DE SÃO CARLOS. Milena Menezes Carvalho. Structural, functional and dynamical properties of a lognormal network of bursting neurons. São Carlos 2017.

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(3) Milena Menezes Carvalho. Structural, functional and dynamical properties of a lognormal network of bursting neurons. Dissertation presented to the Graduate Program in Physics at the Instituto de Física de São Carlos, Universidade de São Paulo, to obtain the degree of Master in Science. Concentration area: Basic Physics Supervisor: Prof. Dr. Leonardo Paulo Maia. Corrected version (Original version available on the Program Unit). São Carlos 2017.

(4) AUTHORIZE THE REPRODUCTION AND DISSEMINATION OF TOTAL OR PARTIAL COPIES OF THIS THESIS, BY CONVENCIONAL OR ELECTRONIC MEDIA FOR STUDY OR RESEARCH PURPOSE, SINCE IT IS REFERENCED.. Cataloguing data reviewed by the Library and Information Service of the IFSC, with information provided by the author Carvalho, Milena Menezes Structural, functional and dynamical properties of a lognormal network of bursting neurons / Milena Menezes Carvalho; advisor Leonardo Paulo Maia reviewed version -- São Carlos 2017. 115 p. Dissertation (Master's degree - Graduate Program in Basic Physics) -- Instituto de Física de São Carlos, Universidade de São Paulo - Brasil , 2017. 1. CA3. 2. Spiking neuron. 3. Spike bursts. 4. Neuronal avalanches. I. Maia, Leonardo Paulo, advisor. II. Title..

(5) To Ledinha.

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(7) ACKNOWLEDGEMENTS. First and foremost, I’d like to thank Dr. Kenji Mizuseki, Dr. Kamran Diba and Dr. György Buzsáki for personally providing additional recordings of the hc3 database and allowing me to use their data to pursue the avalanche studies presented in this dissertation. Especially, I’d like to thank Dr. Kenji Mizuseki for taking his time to help me before and after I received the data hard drive he so benevolently sent me through mail. An entire chapter of this dissertation wouldn’t be without their help. Still in this context, I shouldn’t forget to thank Dr. Tomoki Fukai and Yoshiyuki Omura for allowing me to continue their studies of the MAT neuron network. I arrived at RIKEN in 2015 just as a 3-month trainee, but their trust in my work led me to where I am right now, and to the completition of this dissertation. It wouldn’t be an overstate to say I am only writing these acknowledgements right now because of them, and I’m immensely thankful for that. Acknowledgements wouldn’t be acknowledgements if I didn’t thank my supervisor, my mother, close friends and not-so-close ones for their help throughout the last two (six? twenty four?) years. I have so many people to thank I am even afraid of writing names here, so I won’t. Thank you all for everything. I’ll just add a special thank you to my cats for being the cutest things in the world, because they are. At last, goodbye, Dinah, goodbye. Thanks for all the fish..

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(9) “Though I am often in the depths of misery, there is still calmness, pure harmony and music inside me.” Vicent van Gogh.

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(11) ABSTRACT. CARVALHO, M. M. Structural, functional and dynamical properties of a lognormal network of bursting neurons. 2017. 115p. Dissertation (Master in Science) - Instituto de Física de São Carlos, Universidade de São Paulo, São Carlos, 2017. In hippocampal CA1 and CA3 regions, various properties of neuronal activity follow skewed, lognormal-like distributions, including average firing rates, rate and magnitude of spike bursts, magnitude of population synchrony, and correlations between pre- and postsynaptic spikes. In recent studies, the lognormal features of hippocampal activities were well replicated by a multi-timescale adaptive threshold (MAT) neuron network of lognormally distributed excitatory-to-excitatory synaptic weights, though it remains unknown whether and how other neuronal and network properties can be replicated in this model. Here we implement two additional studies of the same network: first, we further analyze its burstiness properties by identifying and clustering neurons with exceptionally bursty features, once again demonstrating the importance of the lognormal synaptic weight distribution. Second, we characterize dynamical patterns of activity termed neuronal avalanches in in vivo CA3 recordings of behaving rats and in the model network, revealing the similarities and differences between experimental and model avalanche size distributions across the sleep-wake cycle. These results show the comparison between the MAT neuron network and hippocampal readings in a different approach than shown before, providing more insight into the mechanisms behind activity in hippocampal subregions. Keywords: CA3. Spiking neuron. Spike bursts. Neuronal avalanches..

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(13) RESUMO. CARVALHO, M. M. Propriedades estruturais, funcionais e dinâmicas de uma rede lognormal de neurônios bursters. 2017. 115p. Dissertação (Mestrado em Ciências) - Instituto de Física de São Carlos, Universidade de São Paulo, São Carlos, 2017. Nas regiões CA1 e CA3 do hipocampo, várias propriedades da atividade neuronal seguem distribuições assimétricas com características lognormais, incluindo frequência de disparo média, frequência e magnitude de rajadas de disparo (bursts), magnitude da sincronia populacional e correlações entre disparos pré- e pós-sinápticos. Em estudos recentes, as características lognormais das atividades hipocampais foram bem reproduzidas por uma rede de neurônios de limiar adaptativo (multi-timescale adaptive threshold, MAT) com pesos sinápticos entre neurônios excitatórios seguindo uma distribuição lognormal, embora ainda não se saiba se e como outras propriedades neuronais e da rede podem ser replicadas nesse modelo. Nesse trabalho implementamos dois estudos adicionais da mesma rede: primeiramente, analisamos mais a fundo as propriedades dos bursts identificando e agrupando neurônios com capacidade de burst excepcional, mostrando mais uma vez a importância da distribuição lognormal de pesos sinápticos. Em seguida, caracterizamos padrões dinâmicos de atividade chamados avalanches neuronais no modelo e em aquisições in vivo do CA3 de roedores em atividades comportamentais, revelando as semelhanças e diferenças entre as distribuições de tamanho de avalanche através do ciclo sono-vigília. Esses resultados mostram a comparação entre a rede de neurônios MAT e medições hipocampais em uma abordagem diferente da apresentada anteriormente, fornecendo mais percepção acerca dos mecanismos por trás da atividade em subregiões hipocampais. Palavras-chave: CA3. Neurônio disparante. Rajadas de disparo. Avalanches neuronais..

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(15) LIST OF FIGURES. Figure 1 – Neuron functionality schema. Information enters through dendrites, is processed in the soma by emission of an action potential (spike), and exits through the axon via synapses to other neurons. . . . . . . . . . . . . . . .. 31. Figure 2 – Hippocampus curiosities. (a) Human hippocampus and its similarities to a sea horse (“hippocampus” in Greek). (b) Size and form comparison between rat, monkey and human hippocampus. An interesting feature of the hippocampus is its general similarity across species. (c) Rat hippocampus imaging via Golgi’s method (1886). (d) Hippocampal formation subregions and information pathway. (e) CA3 pyramidal cell and its connections. Inset: bursting behavior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 32. Figure 3 – Theoretical neuroscience is still neuroscience. . . . . . . . . . . . . . . . . 33 Figure 4 – Structure of the multi-timescale adaptative threshold (MAT) neuron network. Red circles indicate excitatory (Ex) neurons while blue circles represent inhibitory (In) neurons. Connectivity is random and fixed during the simulation. Left: excitatory-to-excitatory synaptic weights follow a lognormal distribution while others are uniformly distributed (excitatory-to-inhibitory) or fixed (inhibitory-to-excitatory and inhibitory-to-inhibitory). Right: connection probability from excitatory neurons to all neurons is 10%, while inhibitory neurons are highly connected with connection probability 50%.. . 36. Figure 5 – MAT threshold adaptation and firing scheme. When the membrane potential (blue curves) of a neuron reaches its instantaneous threshold value a spike happens and the threshold is increased (red curves). Threshold dynamics allow bursts to appear in an expressive amount without over-activation of the network. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 37. Figure 6 – Raster plot of neuronal activity in the MAT neuron network. Red dots represent activated excitatory neurons spiking at ∼1 Hz; blue dots represent activated inhibitory neurons spiking at ∼15 Hz. . . . . . . . . . . . . . . .. 38. Figure 7 – Distributions of firing rates (top), burst event rates (bottom left) and burst indices (bottom right) for excitatory (in red) and inhibitory neurons (in blue). Lines represent lognormal fits while horizontal bars above the distribution indicate the medians for the first and third quartiles. . . . . . . . . . . . .. Figure 8 – Example of a highly synchronous event in the network (delimited by the shaded area). From top to bottom, raster plot and firing rates of the excitatory (in red) and inhibitory populations (in blue), magnitude of synchrony of the excitatory population, and average membrane potential of two neuron groups with different responses to the synchronous event (one becomes. 39.

(16) hyperpolarized i.e. inhibited, while the other becomes depolarized i.e. excited). 40. Figure 9 – Visualization of how bursts efficiently propagate in the network. Left: schema of a cascade of activity triggered by a spike burst. Activity is tracked for the five strongest synapses in each neuron of the tree. Right: example of an event of burst propagation with time and number of activated neurons as axes. The integers on the right side indicate the depth of the activation tree.. 41. Figure 10 – Illustrative traces of ripples measured in CA1. SPW-R events have been observed across every mammalian species commonly studied in neuroscience, including humans. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 43. Figure 11 – Diversity dependence with list size n (blue curves), determined by equation 3.2. n values are spread from 5 to 10,000 with intervals of 5. The linear fit of highest diversities is f (x) = 1.901 − (4.217 · 10−3 )x (in purple), while lowest diversities are fitted by f (x) = 1.25 − (2.681 · 10−5 )x (in orange). The slope of the curves changes by two orders of magnitude between fits of high and low diversity groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 47. Figure 12 – Scatter plot of the excitatory population using normalized MBS and BC as axes. Top MBS and top BC neurons are identified in red and green respectively, while the rest of the population is pictured in blue. An overlay filter is used to identified the neurons that belong to both top groups, resulting in a yellow hue. Dashed lines are visual indicators of the separation of each group to the overall population. There is an evident branching at (0.4, 0.4), indicating no neuron in this network can show extreme values of MBS and BC at the same time. Neurons with balanced values of normalized MBS and BC don’t show exceptional values for both measurements. At least, two extreme groups of MBS and BC can be defined with around 10 and 30 neurons each, approximately. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 48. Figure 13 – Ranked lists of mean burst size and burst count. Neurons are sorted by descending values of MBS (in red) and BC (in green). Above: the linear fits for the MBS ranked list are, for top neurons, f (x) = 4.63 − 0.253x (in purple), and for intermediate (or typical) neurons, f (x) = 1.647 − (8.204 · 10−5 )x (in orange). Below: in the BC ranked list the linear fits are, for top neurons, f (x) = 156.7 − 3.55x (in purple), and intermediate neurons, f (x) = 25.41 − (5.272 · 10−3 )x (in orange). The linear coefficients of the intermediate fits, 1.647 and 25.41 respectively, were used to define typical values of MBS and BC for the general population. . . . . . . . . . . . . . . . . . . . . . . . .. Figure 14 – Histograms of synaptic weights measurements, (a) mean , (b) standard deviation and (c) maximum, for all neurons in the network (in blue) and for top 100 MBS (in red) and BC neurons (in green). Two distinct situations are shown: neuron as pre- (left column) and post-synaptic (right column).. 49.

(17) Top neurons stand out from the rest of the population for all measurements, especially for maximum weights. Notice how the distribution of maximum weights for the whole population follows a lognormal pattern, as expected from a network with a lognormal rule of excitatory-to-excitatory synaptic weights and 10,000 excitatory neurons. . . . . . . . . . . . . . . . . . . . .. 51. Figure 15 – Density colormap of mean and maximum synaptic weight for the whole population (in blue) and top MBS neurons (in red). Since top MBS list only contained 100 neurons, it would be impossible to reproduce the same density scale of the whole excitatory population with 10,000 neurons. We opted to use a smoothing filter when overlaying both maps for better visualization, and the original ones are shown in the right column (with axis names omitted but identical to the figure on the left). Top MBS neurons are concentrated in the top left corner of the map, while the population as a whole shows a density peak at the center. . . . . . . . . . . . . . . . . . . . . . . . . . .. 52. Figure 16 – Density colormap of mean and maximum synaptic weight for the whole population (in blue) and top BC neurons (in green). The same smooth filtering effect of figure (15) is applied, and the original maps are shown in the right column (with axis names omitted but identical to the figure on the left). The highest density point is in the top right corner of the map, but high density points are scattered around the map in contrast with top MBS neurons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 52. Figure 17 – Time evolution of the membrane potential v (in blue) and synaptic conductance balance g (E) − g (I) (in red) of exceptionally bursty neurons identified as 2825 (top) and 5986 (bottom). Dashed line indicate g (E) − g (I) = 0, balanced excitation and inhibition. 2825 and 5986 are part of both top MBS and BC lists, and are ranked #1 and #2 respectively in the top MBS list (see table (2)). Longer periods of increased excitation over inhibition allow for larger burst sizes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 54. Figure 18 – Time evolution of the membrane potential v (in blue) and synaptic conductance balance g (E) − g (I) (in red) of typically bursty neurons identified as 5101 (top) and 6800 (bottom). Dashed line indicate g (E) − g (I) = 0, balanced excitation and inhibition. 5101 and 6800 are referenced in table (4) and their MBS values are 1.65 and 1.648 respectively. They show single spikes and bursts, but their excitability periods are shorter and excitation and inhibition are more balanced over time. . . . . . . . . . . . . . . . . . . . . . . . . .. Figure 19 – Time evolution of the membrane potential v (in blue), accumulated excitatory (E). synaptic conductance gacc (in red) and strongest synapses participation ratio gratio (in green) of exceptionally bursty neurons identified as 2825 (top) and 5986 (bottom), same neurons and interval as figure (17) for comparison. The. 55.

(18) synaptic weights of the five strongest synapses are so large that they are able to impact v considerably even if they are acting alone (high gratio ). . . . . .. 57. Figure 20 – Time evolution of the membrane potential v (in blue), accumulated excitatory (E). synaptic conductance gacc (in red) and strongest synapses participation ratio gratio (in green) of typically bursty neurons identified as 5101 (top) and 6800 (bottom), the same neurons and interval as figure (18) for comparison. Unlike exceptionally bursty neurons, there are moments of high gratio with low v (E). response as gacc isn’t increased as much. . . . . . . . . . . . . . . . . . . .. 58. Figure 21 – Scatter plot for five‡‡ exceptional (above, in red) and typical (below, in (E). (E). blue) neurons with gacc and gratio as axes. Circles represent (gacc , gratio ) samples measured in the simulation at 1 kHz for all neurons of each category. Differences in concentration suggest the differences the differences in bursting behavior measured in the network. . . . . . . . . . . . . . . . . . . . . . .. 59. Figure 22 – Avalanche definition and examples. Above: by dividing the neuronal activity raster plot into sections of width ∆t, it is possible to identify bins with and without spiking activity (in white and gray, respectively. Neuronal spikes are indicated by blue circles). We call an avalanche the activity between two empty bins and measure its duration (number of non-empty bins times ∆t) and size (spike count during its occurrence, see bottom schema). The interval between two avalanches (number of empty bins times ∆t) defines the interavalanche interval. . . . . . . . . . . . . . . . . . . . . . . . . . .. 64. Figure 23 – Thresholding mechanism. It may be necessary to apply a threshold as shown in red if the network is continuously active. Only the activity above threshold (in saturated blue) will then be used to measure avalanche sizes and durations and interavalanche intervals. . . . . . . . . . . . . . . . . . . . . . . . . .. 64. Figure 24 – Dependence of avalanche distributions, avalanche size (a), duration (b) and IAI (c), with temporal binning for experimental data (left, represented by filled rhombi) and simulations (right, represented by unfilled rhombi). Different colors indicate different temporal bins (i.e., different α). Experimental avalanches were measured from SWS readings of rat ec013, concatenated session ec013.921_927. hIEIi readings show 7.92ms for experimental data and 0.031 ms for simulation. Notice how both experimental and model avalanche size distributions of different α values appear to “collapse” at the same size bin. See annex B for more examples including different animals and vigilance states, and annex C for different simulation trials. . . . . . . . . . . . . . .. Figure 25 – Threshold dependence of avalanche size distributions for model simulations. Above, in blue hues: for α = 1.5, hIEIi = 0.031 ms and thus ∆t = 0.047 ms. The integer median of the activity histogram is 1 and its half was converted to 0, therefore “threshold as half-median” and “no threshold” are the same. 67.

(19) distribution and only one of them is shown (“no threshold”, Θ = 0). Below, in purple hues: for α = 3.0, ∆t = 0.093 ms. The integer median of the activity histogram is 3 and its half was converted to 1. It’s interesting to notice the difference between the distributions tagged as “threshold as half-median” (Θ = 1) and “no threshold” for α = 3.0: a threshold of only 1 was already capable of partially hiding the bump at the tail of the distribution. . . . . .. 69. Figure 26 – Temporal binning and threshold dependence of avalanche size distributions for experimental data (left, represented by filled rhombi) and simulations (right, represented by unfilled rhombi). Red hues indicate α = 1.5 and green hues indicate α = 4.0 and, in both cases, darker colors indicate higher values of threshold. Differently from figure (25), both histograms follow the same fixed log-spaced bins of figure (24) and include the same data.. . . . . . . . 69. Figure 27 – Dependence of relative mean avalanche size s, relative mean duration d and relative branching parameter σ with vigilance state and temporal binning. e measurements for individual session data of rats Lines represent se, de and σ. ec013, gor01 and vvp01 (ec016 was removed for better visualization). Crosses represent actual values of s, d and σ measured over all session files of each vigilance state. Each color represents a vigilance state: SWS in blue, RUNstill in green, REMtonic in red and REMphasic in orange. For all α values, SWS shows relatively larger avalanche sizes and longer durations, with the highest values of relative branching parameter over all categories. . . . . . . . . . .. 72. Figure 28 – Dependence of relative mean avalanche size s (in blue), relative mean duration d (in green) and relative branching parameter σ (in red) with vigilance state. Boxes picture the median of each of the three measurements over all α values for all sessions across all rats for each vigilance state, while error bars indicate the 25th and 75th percentiles. REMtonic is now comparable to SWS because of the particularities of rat ec016. . . . . . . . . . . . . . . . . . . . . . .. 73. Figure 29 – Dependence of relative mean avalanche size s (in blue), relative mean duration d (in green) and relative branching parameter σ (in red) with vigilance state for each rat. Boxes picture the median of each of the three measurements over all α values across all sessions for each vigilance state, while error bars indicate the 25th and 75th percentiles. REMphasic still holds the lowest values of all measurements but each rat shows its particularities with REMtonic measurements. vvp01 and gor01 spiking data were not originally sorted into RUNstill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Figure 30 – Distance measure ζk for number of bins N=29, lognormally spaced. Lines join the median of ζk over all sessions and trials for each vigilance state and simulation, while error bars indicate the 25th and 75th percentiles. Simulation data is represented by purple, while experimental data is pictured as follows:. 74.

(20) SWS in blue, RUNstill in green, REMtonic in red and REMphasic in orange. Except from REMphasic , all other vigilance states have global minima around bins [3.29, 4.17] and [4.17, 5.30]. The same is true for simulations, though its minimum is much more accentuated than its experimental counterparts. . .. 76. Figure 31 – Avalanche size distributions for all α values. Avalanche readings of all sessions of all animals are concatenated into each category (vigilance state). Throughout this section, the color code is fixed: purple for simulation data, blue for SWS, green for RUNstill , red for REMtonic and orange for REMphasic . Visually, low (0.5) and high (3.0, 4.0) values of α show greater differences between simulation and all vigilance states for larger avalanche sizes. . . . .. 78. Figure 32 – Power-law fitting (dashed lines) of avalanche readings of simulation (unfilled rhombi) and vigilance states (filled rhombi). In all graphs, α = 1.0. Color code: simulation data in purple, SWS in blue, RUNstill in green, REMtonic in red and REMphasic in orange.. . . . . . . . . . . . . . . . . . . . . . . . . 80. Figure 33 – Lognormal fitting (filled lines) of avalanche readings of simulation (unfilled rhombi) and vigilance states (filled rhombi). In all graphs, α = 1.0. Color code: simulation data in purple, SWS in blue, RUNstill in green, REMtonic in red and REMphasic in orange.. . . . . . . . . . . . . . . . . . . . . . . . .. 81. Figure 34 – Direct comparison between simulation (unfilled rhombi) and experimental data (filled rhombi) of different vigilance states for all animals (α = 1.0). Filled lines show the lognormal fits introduced in figure (33). Color code: simulation data in purple, SWS in blue, RUNstill in green, REMtonic in red and REMphasic in orange.. . . . . . . . . . . . . . . . . . . . . . . . . . . 83. Figure 35 – Direct comparison between simulation (unfilled rhombi) and experimental data (filled rhombi) of different vigilance states for all animals (α = 2.0). Filled lines show the lognormal fits introduced in figure (33). Simulation and experimental avalanche sizes seem more superposed for this α value, but fitting parameters suggest otherwise. Color code: simulation data in purple, SWS in blue, RUNstill in green, REMtonic in red and REMphasic in orange. .. 84. Figure 36 – Differences across animals concerning avalanche size distributions for SWS (in blue) and REMtonic (in red) (α = 1.0). Rhombi: linearly-spaced binning of avalanche sizes. Filled lines, saturated: lognormally-spaced binning of avalanche sizes. Filled lines, transparent: lognormal fits (figure (33)). . . . .. 86. Figure 37 – Differences across animals when comparing SWS (in blue, filled rhombi) and REMtonic (in red, filled rhombi) avalanche size distributions to simulation (in purple, unfilled rhombi). In all cases, α = 1.0. Filled lines show the lognormal fits introduced in figure (33). . . . . . . . . . . . . . . . . . . . . . . . . .. Figure 38 – Example of avalanche distributions measured for rat vvp01, concatenated session 2006-4-18. Behavioral activities included mazes linearOne, linearTwo,. 87.

(21) Open and Tmaze. Recording duration is approximately 2 hours and 4 electrodes with 8 recording sites each were implanted into CA3. Animal was familiarized with all tasks, having performed each 10 times or more. . . . . 111. Figure 39 – Example of avalanche distributions measured for rat gor01, concatenated session 2006-6-12. Behavioral activities included mazes linearOne, linearTwo and Tmaze. Recording duration is approximately 3,8 hours and 8 electrodes with 8 recording sites each were implanted into CA3. Animal was familiarized with all tasks, having performed each 10 times or more. . . . . . . . . . . . 112. Figure 40 – Example of avalanche distributions measured for rat ec013, concatenated session ec013.931_942. Behavioral activities included mazes linear, bigSquare and wheel. Recording duration is approximately 3,9 hours and 4 electrodes with 8 recording sites each were implanted into CA3. Animal was familiarized with all tasks, having performed each 10 times or more. . . . . . . . . . . . 113. Figure 41 – Example of avalanche distributions measured for rat ec016, concatenated session ec016.914_932. Behavioral activities included mazes linear, bigSquare and wheel. Recording duration is approximately 6,3 hours and 5 electrodes with 8 recording sites each were implanted into CA3. Animal was familiarized with all tasks, having performed each 10 times or more. . . . . . . . . . . . 114. Figure 42 – Example of avalanche distributions measured in different trials of model simulation. Network was simulated for 120 seconds following the parameters set in chapter 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.

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(23) LIST OF TABLES. Table 1 – Terminology openly used in this text. . . . . . . . . . . . . . . . . . . . . . 30 Table 2 – Top ten largest mean burst sizes in the excitatory population. ID is the neuron identification number, BS SD is the standard deviation of burst size and Max BS is the maximum burst size measured. . . . . . . . . . . . . . . . . . . .. 45. Table 3 – Top ten highest burst counts in the excitatory population. ID is the neuron identification number, BS SD is the standard deviation of burst size and Max BS is the maximum burst size measured. . . . . . . . . . . . . . . . . . . .. 46. Table 4 – Intermediate, typical mean burst sizes seed 1000 based on the linear coefficient of intermediate MBS neurons, 1.647, introduced in figure (13). Once again, ID is the neuron identification number, BS SD is the standard deviation of burst size and Max BS is the maximum burst size measured. . . . . . . . . .. 56. Table 5 – Power-law and lognormal fitting parameters for avalanche readings from all sessions of all animals (α = 1.0). . . . . . . . . . . . . . . . . . . . . . . . 79 Table 6 – Power-law and lognormal fitting parameters for avalanche readings from all sessions of all animals (α = 2.0). . . . . . . . . . . . . . . . . . . . . . . . 82 Table 7 – Avalanche count for all sessions of each vigilance state (α = 1.0). As expected, ec016 had the highest counts for all sessions, as it was the animal with the longest total recording hours. . . . . . . . . . . . . . . . . . . . . . . . . .. Table 8 – Power-law and lognormal fitting parameters for avalanche size distributions of rat ec013 (α = 1.0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 9 – Power-law and lognormal fitting parameters for avalanche size distributions of rat ec016 (α = 1.0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 10 – Power-law and lognormal fitting parameters for avalanche size distributions of rat gor01 (α = 1.0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 11 – Power-law and lognormal fitting parameters for avalanche size distributions of rat vvp01 (α = 1.0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 12 – Behavior description. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Table 13 – Number of recorded cells. . . . . . . . . . . . . . . . . . . . . . . . . . . Table 14 – Number of principal cells. . . . . . . . . . . . . . . . . . . . . . . . . . . Table 15 – Number of interneurons. . . . . . . . . . . . . . . . . . . . . . . . . . . Table 16 – Histogram bins: starting and ending points and sizes. These values were fixed. 84. . 85 . 85 . 85 . . . . .. 85 103 104 104 104. across experimental and model avalanche analyses to allow ζk computation. . 109.

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(25) LIST OF ABBREVIATIONS AND ACRONYMS. BOLD. Blood-oxygen-level dependent contrast imaging. BC. Burst count. BS. Burst size. CA. Cornu Ammonis. CDF. Cumulative distribution function. CCDF. Complementary cumulative distribution function. DG. Dentate gyrus. EC. Entorhinal cortex. EEG. Electroencephalogram. EPSP. Excitatory post-synaptic potential. GPU. Graphics processing unit. IAI. Interavalanche interval. IEI. Interevent interval. ISI. Interspike interval. LFP. Local field potential. LIF. Leaky integrate-and-fire. MAT. Multi-timescale adaptive threshold. MBS. Mean burst size. MEG. Magnetoencephalogram. NREM. Non-rapid eye movement sleep. PDF. Probability density function. PGO. Ponto-geniculo-occipital. REM. Rapid eye movement sleep. SOC. Self-organized criticality.

(26) SPW-R. Sharp-wave ripple. SD. Standard deviation. STS. Separation of time scales. SWS. Slow-wave sleep.

(27) LIST OF SYMBOLS. vi. Membrane potential of the i-th neuron. τm. Membrane time constant. RIi. Total input generated by the excitatory and inhibitory synapses of the i-th neuron. VL. Reversal potential of leaky current. VE. Reversal potential of excitatory post-synaptic current. VI. Reversal potential of inhibitory post-synaptic current. θi (t). Adaptive threshold of the i-th neuron. Si. Spike times of the i-th neuron. ω. Threshold resting value. τi,k. Time constant of the k-th threshold component of the i-th neuron. Ai,k. Weight of k-th threshold component of the i-th neuron. gi (t). Synaptic conductance of the i-th neuron. τs. Synaptic decay constant. di,j. Synaptic delay from the i-th to the j-th neuron. Gij. Synaptic weight between the j-th and i-th neurons. δ(t). Dirac’s delta function. µL. Mean of the lognormal distribution of excitatory-to-excitatory synaptic weights. σL. Standard deviation of the lognormal distribution of excitatory-to-excitatory synaptic weights. pfail. Synaptic transmission failure probability. U. Uniqueness operator. L. Length operator. g (E) − g (I). Synaptic conductance balance.

(28) (E) gacc. Accumulated excitatory synaptic conductance. (E,Max) gacc. Accumulated excitatory synaptic conductance originated from the strongest synapses. gratio. Participation percentage of the strongest synapses in the accumulated excitatory synaptic conductance. Sit. Spike of the i-th neuron ocurring at time t. s. Avalanche size. d. Avalanche duration. ∆t. Activity time bin. Θ. Activity threshold for avalanche definition. R. Rate of population activity. h·i. Average in time. α. Multiplication factor of hIEIi. h·iX. Average over X. s. Relative mean avalanche size. se. Relative avalanche size of one session. d. Relative mean avalanche duration. de. Relative avalanche duration of one session. σ0. Single-step branching parameter. σ. Relative branching parameter. σe. Relative branching parameter of one session. ζk. Distance measurement between distributions of different α values for avalanche size bin k. −k. Fitted theoretical power-law distribution slope. µlog. Fitted theoretical lognormal distribution mean. σlog. Fitted theoretical lognormal distribution standard deviation.

(29) CONTENTS. 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29. 1.1. Neuroscience: basic concepts and terminology . . . . . . . . . . . . . 30. 1.2. Hippocampus: structure and function . . . . . . . . . . . . . . . . . . 30. 1.3. On this dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31. 2. NEURON MODEL AND NETWORK DESCRIPTION . . . . . . . . 35. 2.1. Review: a lognormal brain . . . . . . . . . . . . . . . . . . . . . . . . . 35. 2.2. Lognormal recurrent network of MAT neurons . . . . . . . . . . . . . 35. 2.3. Remarkable findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39. 2.4. Chapter conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42. 3. PROPERTIES OF BURSTY NEURONS IN THE MAT NEURON NETWORK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43. 3.1. Review: burstiness in the hippocampus . . . . . . . . . . . . . . . . . 43. 3.2. Burst measurements: mean burst size, burst count . . . . . . . . . . 44. 3.3. Bursty neurons clustering . . . . . . . . . . . . . . . . . . . . . . . . . 48. 3.4. Influence of the strongest synapses . . . . . . . . . . . . . . . . . . . 53. 3.5. Chapter conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60. 4. NEURONAL AVALANCHES IN THE MAT NEURON NETWORK AND IN VIVO RECORDINGS OF RAT CA3 . . . . . . . . . . . . . 61. 4.1. Review: neuronal avalanches and critical phenomena in the brain . . 61. 4.2. Experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62. 4.3. Neuronal avalanche and their measurements . . . . . . . . . . . . . . 62. 4.4. Binning and threshold dependence . . . . . . . . . . . . . . . . . . . . 65. 4.4.1. Temporal binning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65. 4.4.2. Threshold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68. 4.5. Differences across vigilance states . . . . . . . . . . . . . . . . . . . . 70. 4.6. “Collapse” for avalanche size distributions . . . . . . . . . . . . . . . 75. 4.7. Distribution fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76. 4.7.1. Fitting power-laws and lognormals . . . . . . . . . . . . . . . . . . . . . . 77. 4.7.2. Experiment-model comparison . . . . . . . . . . . . . . . . . . . . . . . . 79. 4.7.3. A closer look at animal variability . . . . . . . . . . . . . . . . . . . . . . 82. 4.8. Chapter conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88. 5. CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89.

(30) BIBLIOGRAPHY. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91. APPENDIX APPENDIX A – EXPERIMENTAL DATASET DESCRIPTION. 101 . . 103. APPENDIX B – TWO-SAMPLE KS TEST . . . . . . . . . . . . . 105. ANNEX. 107. ANNEX A – ζk FIXED HISTOGRAM BINS . . . . . . . . . . . . . 109 ANNEX B – EXAMPLE DISTRIBUTIONS OF AVALANCHE SIZE, DURATION AND INTERAVALANCHE INTERVAL: EXPERIMENTAL DATA . . . . . . . . . . . . . . . . 111 ANNEX C – EXAMPLE DISTRIBUTIONS OF AVALANCHE SIZE, DURATION AND INTERAVALANCHE INTERVAL: SIMULATION DATA . . . . . . . . . . . . . . . . . . 115.

(31) 29. 1 INTRODUCTION. Advances in computer science have driven notably the development of neuroscience in the last decades. Countless methods for measuring and manipulating neurons are now available, outputting extremely large data sets that require increasingly complex analysis techniques.1 Signs of progress can be seen not only in experimental data analyses but also in theoretical approaches to understand and solve long-standing puzzles. It is true that the development and consequent experimentation of theoretical models have been in course of several years, but never before has the need for theoretical approaches been so high nor the prospects for advancement so great.1 Theoretical and computational neuroscience has been especially used for studies of the hippocampus, a structure that appears to be crucial to the formation of long-term memories in our brain.2 More robust experiments are being used in search of more comprehensive data both in vitro and in vivo,3 whilst models and simulations are slowly becoming able to reproduce important phenomena of this structure, such as the sharp-wave ripple (SPW-R) events.4 However, contrary to our symmetrybiased common sense, recent findings have shown skewed, non-Gaussian statistics arising in different structures of the brain, for distinct phenomena3, 5, 6 (indeed, SPW-R events are one example7 ). Although these non-trivial statistics are now extensively evidenced by experiments, explaining their origin and their purpose is still an open question in this field, and neuroscientists are actively searching for new theoretical models to accurately reproduce the experimental data.3–5 Findings of heavy-tailed statistical distributions are not exclusive to the hippocampus and the cortex: similar results were found in distinct areas of the brain, for all observation scales. Oscillations measured by electroencephalogram (EEG) on the macroscopic scale, local field potentials (LFP) on the mesoscopic scale and firing rate frequencies of individual neurons on the microscopic scale are some examples.3 These results suggest there is no privileged scale in the brain since microscopic effects can be observed on the macroscopic scale, thus violating the principle of independence along spatial scales.5 Lognormal, power-law and double exponential distributions have been observed experimentally, among others that fit the heavy-tailed distribution category.5 Skewed heavy-tailed distributions are not particular to neuroscience and they have been observed in many scenarios studied by statistical physics, thus uniting neuroscientists and physicists in theoretical studies of the mechanisms that give rise to them..

(32) 30. 1.1. Neuroscience: basic concepts and terminology. Throughout this text, the understanding of many basic concepts of neuroscience will be necessary. Table (1) and figure (1) describe the terms and concepts most used in this text at a superficial level to the unfamiliarized reader. Though this level of description should be sufficient to understand this text, we highly suggest the following references for more in-depth descriptions: KANDEL et al.8 for a biological approach and DAYAN and ABBOTT9 for a more theoretical approach. Table 1 – Terminology openly used in this text. Concept. Description. Neuron. Neurons are the elemental processing units of the brain. These cells are composed of three functionally distinct parts that define the information pathway: dendrites, soma, and axon. Dendrites receive information from other neurons that are processed in the soma. If the input is enough to activate the neuron, an output signal is sent through the axon. Spikes are the indicators of neuronal activation. They are identified in experiments by action potentials that travel across the axon. Spike bursts (or simply bursts) are sequences of spikes that occur in a short time interval. Different neurons show different bursting properties. In this text, a burst is characterized by three or more spikes with intervals shorter than 6 ms. Synapses are the connections between neurons. Through these connections, a neuron can excite or inhibit its neighbors, defining excitatory and inhibitory synapses respectively. The information pathway is described in function of a pre-synaptic neuron, which sends information, and its receiver, the post-synaptic neuron. The signal is modulated by the synaptic weight. Synapses can connect neurons in the same neighborhood or even in different regions of the brain.. Spike Burst. Synapse. Source: By the author.. 1.2. Hippocampus: structure and function. The hippocampus (from Greek, “seahorse”) is part of a subregion of the brain identified as hippocampal formation, which also includes the dentate gyrus, subiculum, presubiculum, parasubiculum and entorhinal cortex; this subregion is particularly interesting as its basic cell and pathway layouts are similar in all mammals11 (see figure (2b)). Alongside with the prefrontal cortex, the hippocampus is crucial for encoding and storing explicit memories, the conscious recall of information.8 While the prefrontal cortex mediates working memories, those stored for short periods of time, the hippocampus is responsible for the formation of long-term memories via activity-dependent synaptic.

(33) 31. Figure 1 – Neuron functionality schema. Information enters through dendrites, is processed in the soma by emission of an action potential (spike), and exits through the axon via synapses to other neurons. Source: IZHIKEVICH10. plasticity.8, 11 Other interesting functions of the hippocampus are spatial navigation and spatial memory, mediated by place cells.8, 11 The hippocampus proper is divided into three subdivisions: Cornu Ammonis 1 (CA1), CA2 and CA3. The information pathway in the hippocampal formation is mostly unidirectional, and information arrive in the hippocampus proper from the dentate gyrus via mossy fibers connected to the pyramidal cells of CA3, which in turn are a major input source to the CA1 hippocampal field via Schaffer collaterals.11 Spike bursts are considered a hallmark feature of CA3 and they can be triggered in many ways; however, the most robust bursts are generated when the entire network becomes synchronously active,11 triggering ripples of activity in CA1 known as sharp-wave ripples (SPW-R).6, 12 In our studies, we were particularly interested in replicating CA3 activity to understand the mechanisms that give rise to SPW-Rs,4 since they are thought to be important for memory consolidation.12–15 1.3. On this dissertation. Large-scale recordings of the hippocampus have provided new insights into the large differences in activities among neurons. In CA1 and CA3, various features of population neuronal activity were shown to obey lognormal-like statistics, including distributions of firing rates, burstiness, and population synchrony.7 Recently, our multi-timescale adaptive threshold (MAT) neuron network was shown to replicate most of the skewed features in spontaneous CA3 activity by introducing lognormally distributed synaptic weights in contrast to a Gaussian distribution.4 Since some quantitative aspects still deviated from those acquired during experiments, the model underwent additional studies for better.

(34) 32. (a). (c). (b). (d). (e). Figure 2 – Hippocampus curiosities. (a) Human hippocampus and its similarities to a sea horse (“hippocampus” in Greek). (b) Size and form comparison between rat, monkey and human hippocampus. An interesting feature of the hippocampus is its general similarity across species. (c) Rat hippocampus imaging via Golgi’s method (1886). (d) Hippocampal formation subregions and information pathway. (e) CA3 pyramidal cell and its connections. Inset: bursting behavior. Source: Adapted from ANDERSEN et al.11. understanding. In this dissertation, we describe two new analyses of the MAT neuron network: first, since spike bursts are thought to be essential for information propagation in our network,4 in chapter 3 we studied and classified classes of bursty neurons that naturally appear in the model as a result of the lognormal distribution of synaptic weights. Additionally, we analyzed the impact of strong synapses in the firing pattern of a neuron, and how they gave rise to exceptional burstiness. Second, we analyzed a different aspect of the dynamics of the network by introducing the concept of avalanches, spatiotemporal patterns of activity proposed in the study of dynamical systems poised at criticality.16 Recent studies suggest that the brain may not work at a critical point, but near it;17, 18 moreover, there is still some discussion on whether neuronal avalanches are universal across brain states.19, 20 In chapter 4, we studied the.

(35) 33. behavior of avalanches in experimental data of rat CA3 for different vigilance states and in our model to verify if the MAT network was able to replicate yet another activity pattern of in vivo data. However, before describing the new analyses, we present the MAT neuron network in chapter 2 as it was introduced in our previous study, including an overview of the interesting results published in OMURA et al.4 Finally, with this dissertation, we wish to bring statistical physics and theoretical neuroscience closer to experimental neuroscience by valuing model-experiment comparison and confrontation, as we personally believe models should ultimately try to reproduce their inspirations, not just exist by themselves (figure (3)).. Figure 3 – Theoretical neuroscience is still neuroscience. Source: IZHIKEVICH10.

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(37) 35. 2 NEURON MODEL AND NETWORK DESCRIPTION. 2.1. Review: a lognormal brain. Recent discoveries of skewed statistical distributions with heavy tails in various structures and scales of the brain, both in vitro and in vivo, have startled the neuroscience community since it isn’t unusual to assume symmetrical, bell-shaped distributions for a large number of brain parameters.3 Even though there’s now an increasing number of evidences of such distributions in experimental data,7, 21–28 little is known about the underlying mechanisms that give rise to them. This breakthrough may help us to better understand the functional and structural organization of the brain in two stages: first, by acquiring more comprehensive and precise data of the systems of interest; and second, by developing and simulating theoretical models in order to extract the necessary and sufficient parameters to reproduce these non-Gaussian statistics. In this chapter, we’ll present our methods for the progress of the latter. Our network was modeled to reproduce the long-tailed, lognormal features of hippocampus activity. The structure of the network developed from the connectivity presented in TERAMAE et al.29 with the addition of a multi-timescale adaptive threshold (MAT) mechanism30 to improve bursting activity. The network achieves a low-frequency spontaneous firing state of bursty neurons that well replicates the observed statistical properties of pyramidal cells in the hippocampus.7, 31 This model has already been published4 and, in the present study, we examined its capabilities further without making considerable changes to its structure and functionality. The following section was reproduced from OMURA et al.4 and adapted when necessary. We used C++ to build and simulate our network and Python for posterior analyses. 2.2. Lognormal recurrent network of MAT neurons. The network consists of 12,000 (10,000 excitatory and 2,000 inhibitory) MAT neurons connected randomly, as shown in figure (4). The MAT neuron model, as presented in KOBAYASHI et al,30 has two parallel dynamics for the membrane potential and spike threshold. The membrane dynamic is the same as that in a conductance-based leaky integrate-and-fire (LIF) neuron, except the membrane potential vi is not reset to the resting potential after spike generation: τm. dvi = −(vi − VL ) + RIi dt (E). (I). RIi = −τm [gi (vi − VE ) + gi (vi − VI )]. (2.1). (2.2).

(38) 36. where the membrane time constant τm is 20 ms for excitatory neurons and 10 ms for inhibitory neurons, and RI is the total input generated by excitatory and inhibitory synaptic inputs. The reversal potentials of leaky, excitatory and inhibitory post-synaptic currents are VL = -70 mV, VE = 0 mV and VI = -80 mV, respectively.. Ex. lognormally distributed. Inh 2,000. Probability density. 10,000. 0.001. 0.01. 0.1. 1. 10. EPSP [mV]. network downscale 10% conn. probability. 50% conn. probability. Figure 4 – Structure of the multi-timescale adaptative threshold (MAT) neuron network. Red circles indicate excitatory (Ex) neurons while blue circles represent inhibitory (In) neurons. Connectivity is random and fixed during the simulation. Left: excitatory-toexcitatory synaptic weights follow a lognormal distribution while others are uniformly distributed (excitatory-to-inhibitory) or fixed (inhibitory-to-excitatory and inhibitoryto-inhibitory). Right: connection probability from excitatory neurons to all neurons is 10%, while inhibitory neurons are highly connected with connection probability 50%. Source: By the author. The adaptive threshold θi (t) has fast (k=1) and slow (k=2) components and obeys θi (t) =. X. H(t − Si ) + ω. (2.3). Ai,k exp (−t/τi,k ). (2.4). Si. Hi (t) =. X k=1,2. where Si are the spike times of the i-th neuron, ω the resting value of the threshold, τi,k the k-th time constant for the i-th neuron, and Ai,k the weight of the k-th component for the i-th neuron. Each neuron generates a spike when its membrane potential reaches the instantaneous value of the spike threshold, and the threshold value is increased according to equations 2.3 and 2.4 at every spike firing, decaying exponentially to the resting value when the neuron does not fire as shown by the red curves in figure (5). The total synaptic input to the neuron is interrupted during a refractory period of 1 ms. In excitatory neurons, the magnitudes of the fast component Ai,1 are drawn randomly from a normal distribution with.

(39) 37. mean 1.5 mV and standard deviation 0.25 mV, and the magnitude of the slow component is set as Ai,2 = 0.5 mV. In inhibitory neurons, the magnitudes of fast and slow components are set as Ai,1 = 3 mV and Ai,2 = 0 (i.e., the slow component is absent). Excitatory and inhibitory neurons have identical time constants for the fast component, τi,1 = 10 ms, and excitatory neurons have τi,2 = 200 ms for the slow component. For both neuronal types, ω = -55 mV.. Figure 5 – MAT threshold adaptation and firing scheme. When the membrane potential (blue curves) of a neuron reaches its instantaneous threshold value a spike happens and the threshold is increased (red curves). Threshold dynamics allow bursts to appear in an expressive amount without over-activation of the network. Source: By the author. The time course of the synaptic conductance g(t) of the i-th neuron is described as. dgi gi X X =− + Gij δ(t − Sj − di,j ) dt τs j Sj. (2.5). where τs = 2 ms is the decay constant, di,j the synaptic delay from the i-th to the j-th neuron, Gij the weight of the synaptic connection between the j-th and i-th neurons, and δ(t) is Dirac’s delta function. The delays of the excitatory-to-excitatory synaptic connections are uniformly distributed in the range of 1 to 3 ms, and the delays of the other connections have an identical value of 1 ms. Setting the different delays enhances the stability of spontaneous activity.29 Inspired by recent findings of lognormal excitatory post-synaptic conductances in rat hippocampal CA3 pyramidal cells,27 the weights of the excitatory-to-excitatory synaptic connections are drawn from the following lognormal.

(40) 38. distribution independently for individual neurons: . . 1 (ln x − µL )2  Ψ(x) = √ exp  − 2σL2 2πσL x. (2.6). where σL = 1.0 and µL = ln(0.2) + σL2 ≈ −0.61 are the standard deviation and mean of the variable’s natural logarithm (ln x). Equation 2.6 also mimics the typical EPSP amplitude distributions observed in experiments in neocortex.32, 33 Here, any unrealistic value that is greater than 20 mV is avoided by drawing a new value from the distribution. The weights of excitatory-to-inhibitory, inhibitory-to-excitatory, and inhibitory-to-inhibitory synaptic connections are uniform for mathematical simplicity as the corresponding experimental distributions are not known well, and their values are fixed at 0.018, 0.0035, and 0.0025, respectively, to ensure stable spontaneous activity.4 Synaptic transmission fails at excitatory-to-excitatory synapses at a rate depending on the amplitude of the excitatory post-synaptic potential (EPSP): pfail =. 0.1 0.1 + EPSP [mV]. (2.7). The network is randomly connected so that the innervation probabilities of excitatory and inhibitory neurons are 10% and 50%, respectively, and each neuron receives on average 1000 excitatory and 1000 inhibitory synaptic inputs (see figure (4)). Simulations were run for 120 s of spontaneous, irregular activity (as shown in figure (6)) after initialization via Poisson spike trains for 100 ms applied to all neurons at 10 Hz.. Figure 6 – Raster plot of neuronal activity in the MAT neuron network. Red dots represent activated excitatory neurons spiking at ∼1 Hz; blue dots represent activated inhibitory neurons spiking at ∼15 Hz. Source: By the author.

(41) 39. 2.3. Remarkable findings. • Burstiness: Many features of hippocampal bursting31 were reproduced by this model. Model neurons show single spikes and bursts, with typical interspike intervals (ISI) of ∼1-2 ms during bursts, slightly shorter than experimental measurements (∼ 2-4 ms). Some pyramidal cells are more prone to display longer bursts than others, and such variability was also found in our model. Mean firing rate, burst event rate and burst index distributions display lognormal patterns both in vivo (regardless of brain states7 ) and simulation-wise (see figure (7)); however, the lack of greater heterogeneity in the excitatory population of the model doesn’t allow considerable cell-to-cell variability of spiking patterns.. Figure 7 – Distributions of firing rates (top), burst event rates (bottom left) and burst indices (bottom right) for excitatory (in red) and inhibitory neurons (in blue). Lines represent lognormal fits while horizontal bars above the distribution indicate the medians for the first and third quartiles. Source: Adapted from OMURA et al.4. • Synchrony: Population synchrony refers to the synchronous activation of neurons in a time window (see figure (8)). The average interval between highly synchronous events was ∼1.25 Hz, in agreement with intervals of spontaneous sharp-waves in CA3 in vitro (∼1.1–1.6 Hz)34, 35 and in vivo (0.01–2.0 Hz),36 and also with sharp-wave ripple events (SPW-R) in CA1, which are induced by CA3 synchronous activity and occur at average frequencies of 0.3–1.0 Hz during sleep.37, 38 Skewed firing patterns of individual neurons during population synchrony are also in agreement.

(42) 40. with experimental findings,7 although the proportion of spikes in highly synchronous events is positively correlated to the firing rates of individual neurons, contrary to what was measured in vivo. Additionally, the neurons are not uniformly activated during a synchronous event, and excitatory neurons undergo stronger competition during epochs of population synchrony.. Figure 8 – Example of a highly synchronous event in the network (delimited by the shaded area). From top to bottom, raster plot and firing rates of the excitatory (in red) and inhibitory populations (in blue), magnitude of synchrony of the excitatory population, and average membrane potential of two neuron groups with different responses to the synchronous event (one becomes hyperpolarized i.e. inhibited, while the other becomes depolarized i.e. excited). Source: Adapted from OMURA et al.4. • Propagation of bursts: Bursts affect post-synaptic neurons considerably more than single spikes.39 In fact, our model shows that while single spikes only travel a few links along a few pathways of the cascading network of connections of a neuron, bursts travel much longer distances along many more pathways, arriving at a greater number of neighbors (see figure (9)). Additionally, sequences with 3 or 7 activated “children” neurons of a highly bursty “parent” were shown to be more statistically significant than permutated and “imaginary” sequences (i.e., sequences containing.

(43) 41. activity of the children neurons in arbitrary time periods temporally far from any activity from the parent neuron).. Figure 9 – Visualization of how bursts efficiently propagate in the network. Left: schema of a cascade of activity triggered by a spike burst. Activity is tracked for the five strongest synapses in each neuron of the tree. Right: example of an event of burst propagation with time and number of activated neurons as axes. The integers on the right side indicate the depth of the activation tree. Source: Adapted from OMURA et al.4. • Comparison with a Gaussian distribution: When testing a Gaussian distribution of excitatory-to-excitatory synaptic weights, considerable spontaneous activity was only found for a very small range of parameters, and yet in this range activity died incredibly faster. Bursting activity was very limited and exceptionally bursty neurons were not found. All other special properties of the lognormal network were also absent in the Gaussian network..

(44) 42. 2.4. Chapter conclusions. • Our network shows great prospects of CA3 modeling. A combination of a lognormal distribution of excitatory-to-excitatory synaptic weights and a simple neuron model with bursting capability (MAT model) was able to give rise to the heavy-tailed distributions of spike firing, burstiness and population synchrony measured in vivo7 while maintaining a biologically plausible low-frequency spontaneous activity regime. Additionally, the network model accounts for various properties of hippocampal synchronous firing typically observed during SPW-Rs, thus serving as a study tool of the yet unknown underlying mechanisms of these events crucial to memory consolidation. • Bursts are essential. More importantly, our findings showcase the importance of bursts in information propagation. Bursts are more efficiently broadcast over the connectivity tree of a neuron via strong synaptic connections, and some sequences of activated neurons are more statistically significant than others, indicating routes of information transmission..

(45) 43. 3 PROPERTIES OF BURSTY NEURONS IN THE MAT NEURON NETWORK. 3.1. Review: burstiness in the hippocampus. Hippocampal pyramidal neurons in vivo exhibit both single spikes and complex spike bursts,31, 40 and the latter are frequently observed during sharp-wave ripple events (SPW-R).41–43 SPW-Rs, illustrated by figure (10), are transient patterns of activity in the CA1 region of the hippocampus, occurring during slow-wave sleep (SWS) and non-exploratory wake states (that is, states that do not introduce new objects and environments, also known as off-line states)12 ; they are the population pattern of highest synchrony in the mammalian brain and its output affects a wide area of the cortex and several subcortical nuclei.6 Various studies suggest that SPW-Rs are involved in off-line memory consolidation processes.12–15 Moreover, it is known that SPW-R events result from excitation originated by the synchronous activity of subsets of the CA3 population in a population burst.6, 12 It would be therefore interesting to unveil how this spontaneous synchrony arises (taking up to 10-20% of the CA3 population)12, 44 or, in other words, which parameters are strictly necessary to the occurrence of these bursts.. Figure 10 – Illustrative traces of ripples measured in CA1. SPW-R events have been observed across every mammalian species commonly studied in neuroscience, including humans. Source: BUZSÁKI6. As discussed in chapter 2, our MAT neuron network with lognormal connections wellreproduced many characteristics of hippocampal (especially CA3) activity, including the lognormal distributions of mean firing rate and burstiness across the sleep-wake cycle and different behavioral activities.7 This suggests a lognormal distribution of excitatory weights associated with a neuron model with bursting capability could explain the spontaneous population activity in CA3; however, some characteristics measured by experiments weren’t reproduced by the model, such as cell-to-cell variability in patterns of spike firing.45 In.

(46) 44. fact, our previous analyses4 showed no clear border between the two categories of bursty neurons, typical and exceptional bursters∗ , also present in experimental observations of the hippocampus.31 Our efforts to characterize the burstiness in our model network are described in this chapter. We studied how burst measurements are related to synaptic weight inputs while trying to define the class of exceptionally bursty neurons. Additionally, we studied the differences between neurons with large mean burst sizes (MBS) and neurons with high burst counts (BC) to discuss which measurement better describes the burstiness of a neuron. Finally, we also checked how the strongest synapses of a neuron influence its bursting pattern, and how this relates to the existence of different bursty neuron categories.. 3.2. Burst measurements: mean burst size, burst count. In order to characterize bursty neurons in our model we have previously used burst frequency and burst index (the ratio of spikes in bursts to all spikes), and they showed lognormal statistics for excitatory neurons4 as expected from experiments.7 In this study, however, we have redefined them to improve clarity, although the previous and current definitions are still equivalent. In both scenarios, a spike burst is defined as a series of three or more spikes with interspike intervals (ISI) shorter than 6 ms.7 We define the mean burst size (MBS) of a neuron as its average activity size:. MBS =. P. activity. P. size. activity. (3.1). where activity is any spiking event of a neuron and activity size is the number of spikes present in this event. Although we named this measurement mean burst size, single spikes still count as activities of size 1. When a neuron shows bursts but its events are mostly single spikes, its MBS value will drop: MBS has a minimum value of 1, true for neurons that are pure single spikers. It is equivalent to burst index, but additionally to determining whether a great portion of the spikes are generated in bursts, it also conveys an overview of the magnitude of the bursts. Moreover, since burst frequencies are usually lower than 1 Hz, we opted to use burst count instead of frequency to improve clarity, but there is no difference between the two measurements considering all simulations had equal durations. Contrary to MBS, only “real” bursts are considered for BC (three or more spikes, ISI<6 ms). The network was simulated for 120 s using the same parameters originally proposed ∗. Typical bursters show an exponential decay of the probability of generating a burst of length n as a function of n; on the other hand, exceptional bursters show superexponential decay..

(47) 45. in OMURA et al.4 Though stochasticity regulates the initial activation of the network† , variability between different trials was low, and hence we only show here data acquired from one simulation trial. Because of their long durations in comparison to the network timescale, each simulation trial gives us enough information by itself. Tables (2) and (3) show bursting statistics for the top ten excitatory neurons‡ with the largest MBS and highest BC, respectively. MBS reaches values as high as 4.99 and neurons with large MBS display bursts of sizes up to 11. It’s noticeable how rapidly MBS drops from the first to the tenth neuron in the list (∼45%), as does the standard deviation of the measurement (∼46%), and yet BC doesn’t change as much (∼25%). On the other hand, for highest BC, MBS variability is lower across all ten neurons (∼21%) while BC values show a consistent drop across ranks (∼34%), and their maximum burst sizes are generally lower than the top ten MBS neurons. Table 2 – Top ten largest mean burst sizes in the excitatory population. ID is the neuron identification number, BS SD is the standard deviation of burst size and Max BS is the maximum burst size measured. ID. MBS. BS SD. BC. Max BS. 2998 5986 9314 2825 1773 8259 3912 2768 9039 2461. 4.99 4.83 3.82 3.61 3.20 3.12 3.06 2.88 2.74 2.70. 3.02 3.02 3.19 2.73 2.53 3.10 1.98 2.25 1.51 1.61. 68 63 65 52 54 65 64 62 60 51. 10 10 10 9 9 11 7 8 6 6. Source: By the author. This preliminary analysis already raises a question: which definition is the best to define an exceptionally bursty neuron? Should an exceptionally bursty neuron be one that consistently displays long bursts, though they may not be numerous, or a neuron that consistently shows bursts during the recording, though these bursts may not be as long? In OMURA et al.,4 the neurons we referred to as exceptionally bursty had superexponential, non-zero probability of generating bursts of sizes 6 or 7, and this stays true to both groups we described here. Since the answer to the question above still wasn’t clear, both groups were studied further. Nevertheless, as both groups were expanded from ten to a hundred † ‡. We applied Poisson spike trains to all neurons at 10 Hz to initiate activity. After 100 ms, all external input was shut down and the network continued spontaneously active for 120 s. In this chapter we only discuss excitatory neurons since we believed it was the lognormal distribution of excitatory-to-excitatory synaptic weights that generated all of the interesting phenomena we were interested in..

(48) 46 Table 3 – Top ten highest burst counts in the excitatory population. ID is the neuron identification number, BS SD is the standard deviation of burst size and Max BS is the maximum burst size measured. ID. MBS. BS SD. BC. Max BS. 2982 4191 3919 4797 5417 7402 4796 6754 523 7132. 2.20 2.28 2.19 2.28 2.32 1.95 1.83 2.29 2.32 1.83. 1.19 1.32 1.24 1.19 1.21 1.05 1.03 1.22 1.48 1.03. 186 152 145 143 135 133 130 127 125 123. 7 6 5 6 6 5 6 7 8 6. Source: By the author. neurons (1% of the excitatory population), a significant overlap between the two lists was observed, that is, many neurons showed both MBS among the largest and BC among the highest simultaneously. To quantify this overlap we introduced a diversity measurement:. diversity(n) =. L[U[{topMBS}(n) + {topBC}(n)]] n. (3.2). where {topMBS}(n) and {topBC}(n) are respectively the lists of largest MBS and highest BC of size n, U is an uniqueness operator and L is a length operator.§ In short, diversity(n) measures if the two lists of size n are identical (diversity(n) = 1) or completely unique (diversity(n) = 2). We calculated diversity(n) from n = 5 up to n = 10, 000 (all excitatory neurons) with intervals of size 5. Diversity dependence with n is displayed in figure (11). There is a prominent change in behavior in the curve: for small n, both lists are quite unique among themselves and there’s a noteworthy diversity variability across consecutive values of n (large linear slope, evidenced by the linear fit in purple); for large n, the similarity between the lists increase but the diversity variability across n drops (small linear slope, in orange). Since the slope of the first linear approximation is quite steep, for small values of n there is already a substantial overlap between the two lists (at n = 100, 46%). In fact, when drawing a scatter plot of all neurons using normalized MBS and BC as axes (figure (12)), top MBS (in red) and top BC (in green) groups with n = 100 are easily separated from the rest of the population, and the same is valid the overlapping group, indicated by the overlay of §. U, uniqueness operator, transforms a collection of units that can be repeated into a set of only unique, distinct entries. L, length operator, returns the size of the list..

(49) 47. red and green circles¶ . Considering how almost half of the neurons belong to both lists at n = 100, a low n value for a population of 10,000, we could either choose to pick only one measurement to analyze thoroughly or define those overlapping neurons as the real exceptionally bursty neurons. We argued the best description of an exceptional burster should be a neuron that not only has a high burst frequency (high BC) but also displays long bursts including many spikes (large MBS), as combining both effects should maximize bursting advantages of information transmission.4 Consequently, the overlapping neurons should be our best candidates for exceptional burstiness; in any case, there was still some interest in checking what was substantially different about the two lists.. [0:2000] zoom. Figure 11 – Diversity dependence with list size n (blue curves), determined by equation 3.2. n values are spread from 5 to 10,000 with intervals of 5. The linear fit of highest diversities is f (x) = 1.901 − (4.217 · 10−3 )x (in purple), while lowest diversities are fitted by f (x) = 1.25 − (2.681 · 10−5 )x (in orange). The slope of the curves changes by two orders of magnitude between fits of high and low diversity groups. Source: By the author. Finally, inspired by the diversity measurement we also studied how MBS and BC varied in their own ranked lists as shown in figure (13). The curves behave similarly to diversity: their highest values are low in number but great in magnitude, generating the sharp slopes we see in the linear fits (purple lines). This is also a characteristic feature of the lognormal distributions measured previously for burst event rate and burst index in this network.4 In these plots, instead of fitting the curve linearly for lowest ranks (orange lines), we chose to fit the intermediate region, populated by neurons that clearly do not participate in the highest rank lists but are still quite active in the network. By doing so, we were trying to define a group of typically bursty neurons to compare the exceptional groups to. This analysis will be discussed in section 3.4. ¶. An interesting feature of figure (12) is the ramification of the scatter plot into two directions. Though its implications are not yet known, the figure suggests no neuron in this network can have extreme values of MBS and BC at the same time, and consequently, there are at least two clusters of neurons with different properties..

(50) 48. (a) Large mean burst size. (b) High burst count. (c) Top neurons lists combined Figure 12 – Scatter plot of the excitatory population using normalized MBS and BC as axes. Top MBS and top BC neurons are identified in red and green respectively, while the rest of the population is pictured in blue. An overlay filter is used to identified the neurons that belong to both top groups, resulting in a yellow hue. Dashed lines are visual indicators of the separation of each group to the overall population. There is an evident branching at (0.4, 0.4), indicating no neuron in this network can show extreme values of MBS and BC at the same time. Neurons with balanced values of normalized MBS and BC don’t show exceptional values for both measurements. At least, two extreme groups of MBS and BC can be defined with around 10 and 30 neurons each, approximately. Source: By the author. 3.3. Bursty neurons clustering. Knowing which neurons were particularly bursty, the next step was to look at structural differences between them and the rest of the network. As basically all structural variability between neurons come from the excitatory-to-excitatory synaptic weights, we studied how different were the connections of the top 100 MBS and top 100 BC neurons compared to the population as a whole. For all of its excitatory and inhibitory connections, a neuron can be termed as.

(51) 49. Figure 13 – Ranked lists of mean burst size and burst count. Neurons are sorted by descending values of MBS (in red) and BC (in green). Above: the linear fits for the MBS ranked list are, for top neurons, f (x) = 4.63 − 0.253x (in purple), and for intermediate (or typical) neurons, f (x) = 1.647 − (8.204 · 10−5 )x (in orange). Below: in the BC ranked list the linear fits are, for top neurons, f (x) = 156.7 − 3.55x (in purple), and intermediate neurons, f (x) = 25.41 − (5.272 · 10−3 )x (in orange). The linear coefficients of the intermediate fits, 1.647 and 25.41 respectively, were used to define typical values of MBS and BC for the general population. Source: By the author. pre-synaptic when it receives input from a “neighbor”, or post-synaptic when it sends output after activation. The inputs of a neuron will directly affect its firing pattern, while its outputs will affect the firing of others. According to this, if some neurons are exceptionally bursty in a network that is otherwise homogeneous, there must be something different about its inputs, i.e., about the weights that arose from the lognormal distribution. In view of these arguments, we constructed histograms of the mean excitatory synaptic weights, their standard deviation and maximum for all excitatory neurons, and separate histograms of the same measurements for the top 100 MBC and top 100 BC neurons, as shown in figure (14). Excitatory-to-excitatory connectivity is fixed at 10% and hence all excitatory neurons have the same amount of inward (neuron is post-synaptic) and outward (neuron is pre-synaptic) excitatory connections. To highlight the uniqueness of the inward connections in a bursty neuron, we show all three aforementioned measurements in two different contexts: firstly, when the neuron is pre-synaptic (left column), and secondly, when the neuron is post-synaptic (right column). There is a striking difference in the behavior of the histograms for top MBC (in red) and top BC (in green) across the two columns: when a top neuron is pre-synaptic, there’s nothing special about its histograms.