Statistical physics approach to dendritic computation: the excitable-wave mean-field approximation

Brain-wide dendrites in a near-optimal performance of dynamic range and information transmission

Dendrites receive and process signals from other neurons. The range of signal intensities that can be robustly distinguished by dendrites is quantified by the dynamic range. We investigate the dynamic range and information transmission efficiency of dendrites in relation to dendritic morphology. We model dendrites in a neuron as multiple excitable binary trees connected to the soma where each node in a tree can be excited by external stimulus or by receiving signals transmitted from adjacent excited nodes. It has been known that larger dendritic trees have a higher dynamic range. We show that for dendritic tress of the same number of nodes, the dynamic range increases with the number of somatic branches and decreases with the asymmetry of dendrites, and the information transmission is more efficient for dendrites with more somatic branches. Moreover, our simulated data suggest that there is an exponential association (decay resp.) of overall relative energy consumption (dynamic range resp.) in relation to the number of somatic branches. This indicates that further increasing the number of somatic branches (e.g. beyond 10 somatic branches) has limited ability to improve the transmission efficiency. With brain-wide neuron digital reconstructions of the pyramidal cells, 90% of neurons have no more than 10 dendrites. These suggest that actual brain-wide dendritic morphology is near optimal in terms of both dynamic range and information transmission.

Information processing in tree networks of excitable elements

We study the collective response of small random tree networks of diffusively coupled excitable elements to stimuli applied to leaf nodes. Such networks model the morphology of certain sensory neurons that possess branched myelinated dendrites with excitable nodes of Ranvier at every branch point and at leaf nodes. Leaf nodes receive random inputs along with a stimulus and initiate action potentials that propagate through the tree. We quantify the collective response registered at the central node using mutual information. We show that in the strong-coupling limit, the statistics of the number of nodes and leaves determines the mutual information. At the same time, the collective response is insensitive to particular node connectivity and distribution of stimulus over leaf nodes. However, for intermediate coupling, the mutual information may strongly depend on the stimulus distribution among leaf nodes. We identify a mechanism behind the competition of leaf nodes that leads to nonmonotonous dependence of mutual information on coupling strength. We show that a localized stimulus given to a tree branch can be occluded by the background firing of unstimulated branches, thus suppressing mutual information. Nonetheless, the mutual information can be enhanced by a proper stimulus localization and tuning of coupling strength.

Single-neuron dynamical effects of dendritic pruning implicated in aging and neurodegeneration: towards a measure of neuronal reserve

Aging is a main risk factor for neurodegenerative disorders including Alzheimer's disease. It is often accompanied by reduced cognitive functions, gray-matter volume, and dendritic integrity. Although age-related brain structural changes have been observed across multiple scales, their functional implications remain largely unknown. Here we simulate the aging effects on neuronal morphology as dendritic pruning and characterize its dynamical implications. Utilizing a detailed computational modeling approach, we simulate the dynamics of digitally reconstructed neurons obtained from Neuromorpho.org. We show that dendritic pruning affects neuronal integrity: firing rate is reduced, causing a reduction in energy consumption, energy efficiency, and dynamic range. Pruned neurons require less energy but their function is often impaired, which can explain the diminished ability to distinguish between similar experiences (pattern separation) in older people. Our measures indicate that the resilience of neuronal dynamics is neuron-specific, heterogeneous, and strongly affected by dendritic topology and the position of the soma. Based on the emergent neuronal dynamics, we propose to classify the effects of dendritic deterioration, and put forward a topological measure of “neuronal reserve” that quantifies the resilience of neuronal dynamics to dendritic pruning. Moreover, our findings suggest that increasing dendritic excitability could partially mitigate the dynamical effects of aging.

Spatially resolved dendritic integration: towards a functional classification of neurons

The vast tree-like dendritic structure of neurons allows them to receive and integrate input from many neurons. A wide variety of neuronal morphologies exist, however, their role in dendritic integration, and how it shapes the response of the neuron, is not yet fully understood. Here, we study the evolution and interactions of dendritic spikes in excitable neurons with complex real branch structures. We focus on dozens of digitally reconstructed illustrative neurons from the online repository NeuroMorpho.org, which contains over 130,000 neurons. Yet, our methods can be promptly extended to any other neuron. This approach allows us to estimate and map specific and heterogeneous patterns of activity observed across extensive dendritic trees with thousands of compartments. We propose a classification of neurons based on the location of the soma (centrality) and the number of branches connected to the soma. These are key topological factors in determining the neuron's energy consumption, firing rate, and the dynamic range, which quantifies the range in synaptic input rate that can be reliably encoded by the neuron's firing rate. Moreover, we find that bifurcations, the structural building blocks of complex dendrites, play a major role in increasing the dynamic range of neurons. Our results provide a better understanding of the effects of neuronal morphology in the diversity of neuronal dynamics and function.

New Types of Experiments Reveal that a Neuron Functions as Multiple Independent Threshold Units

Neurons are the computational elements that compose the brain and their fundamental principles of activity are known for decades. According to the long-lasting computational scheme, each neuron sums the incoming electrical signals via its dendrites and when the membrane potential reaches a certain threshold the neuron typically generates a spike to its axon. Here we present three types of experiments, using neuronal cultures, indicating that each neuron functions as a collection of independent threshold units. The neuron is anisotropically activated following the origin of the arriving signals to the membrane, via its dendritic trees. The first type of experiments demonstrates that a single neuron’s spike waveform typically varies as a function of the stimulation location. The second type reveals that spatial summation is absent for extracellular stimulations from different directions. The third type indicates that spatial summation and subtraction are not achieved when combining intra- and extra- cellular stimulations, as well as for nonlocal time interference, where the precise timings of the stimulations are irrelevant. Results call to re-examine neuronal functionalities beyond the traditional framework, and the advanced computational capabilities and dynamical properties of such complex systems.

Coexistence of critical sensitivity and subcritical specificity can yield optimal population coding

The vicinity of phase transitions selectively amplifies weak stimuli, yielding optimal sensitivity to distinguish external input. Along with this enhanced sensitivity, enhanced levels of fluctuations at criticality reduce the specificity of the response. Given that the specificity of the response is largely compromised when the sensitivity is maximal, the overall benefit of criticality for signal processing remains questionable. Here, it is shown that this impasse can be solved by heterogeneous systems incorporating functional diversity, in which critical and subcritical components coexist. The subnetwork of critical elements has optimal sensitivity, and the subnetwork of subcritical elements has enhanced specificity. Combining segregated features extracted from the different subgroups, the resulting collective response can maximize the trade-off between sensitivity and specificity measured by the dynamic-range-to-noise ratio. Although numerous benefits can be observed when the entire system is critical, our results highlight that optimal performance is obtained when only a small subset of the system is at criticality.

Approximate-master-equation approach for the Kinouchi-Copelli neural model on networks

In this work, we use the approximate-master-equation approach to study the dynamics of the Kinouchi-Copelli neural model on various networks. By categorizing each neuron in terms of its state and also the states of its neighbors, we are able to uncover how the coupled system evolves with respective to time by directly solving a set of ordinary differential equations. In particular, we can easily calculate the statistical properties of the time evolution of the network instantaneous response, the network response curve, the dynamic range, and the critical point in the framework of the approximate-master-equation approach. The possible usage of the proposed theoretical approach to other spreading phenomena is briefly discussed.

Diversity improves performance in excitable networks

As few real systems comprise indistinguishable units, diversity is a hallmark of nature. Diversity among interacting units shapes properties of collective behavior such as synchronization and information transmission. However, the benefits of diversity on information processing at the edge of a phase transition, ordinarily assumed to emerge from identical elements, remain largely unexplored. Analyzing a general model of excitable systems with heterogeneous excitability, we find that diversity can greatly enhance optimal performance (by two orders of magnitude) when distinguishing incoming inputs. Heterogeneous systems possess a subset of specialized elements whose capability greatly exceeds that of the nonspecialized elements. We also find that diversity can yield multiple percolation, with performance optimized at tricriticality. Our results are robust in specific and more realistic neuronal systems comprising a combination of excitatory and inhibitory units, and indicate that diversity-induced amplification can be harnessed by neuronal systems for evaluating stimulus intensities.

Single-neuron criticality optimizes analog dendritic computation

Active dendritic branchlets enable the propagation of dendritic spikes, whose computational functions remain an open question. Here we propose a concrete function to the active channels in large dendritic trees. Modelling the input-output response of large active dendritic arbors subjected to complex spatio-temporal inputs and exhibiting non-stereotyped dendritic spikes, we find that the dendritic arbor can undergo a continuous phase transition from a quiescent to an active state, thereby exhibiting spontaneous and self-sustained localized activity as suggested by experiments. Analogously to the critical brain hypothesis, which states that neuronal networks self-organize near criticality to take advantage of its specific properties, here we propose that neurons with large dendritic arbors optimize their capacity to distinguish incoming stimuli at the critical state. We suggest that "computation at the edge of a phase transition" is more compatible with the view that dendritic arbors perform an analog rather than a digital dendritic computation.

Scaling behavior in probabilistic neuronal cellular automata

We study a neural network model of interacting stochastic discrete two-state cellular automata on a regular lattice. The system is externally tuned to a critical point which varies with the degree of stochasticity (or the effective temperature). There are avalanches of neuronal activity, namely, spatially and temporally contiguous sites of activity; a detailed numerical study of these activity avalanches is presented, and single, joint, and marginal probability distributions are computed. At the critical point, we find that the scaling exponents for the variables are in good agreement with a mean-field theory.

Dynamic range of vertebrate retina ganglion cells: importance of active dendrites and coupling by electrical synapses

The vertebrate retina has a very high dynamic range. This is due to the concerted action of its diverse cell types. Ganglion cells, which are the output cells of the retina, have to preserve this high dynamic range to convey it to higher brain areas. Experimental evidence shows that the firing response of ganglion cells is strongly correlated with their total dendritic area and only weakly correlated with their dendritic branching complexity. On the other hand, theoretical studies with simple neuron models claim that active and large dendritic trees enhance the dynamic range of single neurons. Theoretical models also claim that electrical coupling between ganglion cells via gap junctions enhances their collective dynamic range. In this work we use morphologically reconstructed multi-compartmental ganglion cell models to perform two studies. In the first study we investigate the relationship between single ganglion cell dynamic range and number of dendritic branches/total dendritic area for both active and passive dendrites. Our results support the claim that large and active dendrites enhance the dynamic range of a single ganglion cell and show that total dendritic area has stronger correlation with dynamic range than with number of dendritic branches. In the second study we investigate the dynamic range of a square array of ganglion cells with passive or active dendritic trees coupled with each other via dendrodendritic gap junctions. Our results suggest that electrical coupling between active dendritic trees enhances the dynamic range of the ganglion cell array in comparison with both the uncoupled case and the coupled case with cells with passive dendrites. The results from our detailed computational modeling studies suggest that the key properties of the ganglion cells that endow them with a large dynamic range are large and active dendritic trees and electrical coupling via gap junctions.

Signal integration enhances the dynamic range in neuronal systems

The dynamic range measures the capacity of a system to discriminate the intensity of an external stimulus. Such an ability is fundamental for living beings to survive: to leverage resources and to avoid danger. Consequently, the larger is the dynamic range, the greater is the probability of survival. We investigate how the integration of different input signals affects the dynamic range, and in general the collective behavior of a network of excitable units. By means of numerical simulations and a mean-field approach, we explore the nonequilibrium phase transition in the presence of integration. We show that the firing rate in random and scale-free networks undergoes a discontinuous phase transition depending on both the integration time and the density of integrator units. Moreover, in the presence of external stimuli, we find that a system of excitable integrator units operating in a bistable regime largely enhances its dynamic range.