Democratization in a passive dendritic tree: an analytical investigation

One way to achieve amplification of distal synaptic inputs on a dendritic tree is to scale the amplitude and/or duration of the synaptic conductance with its distance from the soma. This is an example of what is often referred to as “dendritic democracy”. Although well studied experimentally, to date this phenomenon has not been thoroughly explored from a mathematical perspective. In this paper we adopt a passive model of a dendritic tree with distributed excitatory synaptic conductances and analyze a number of key measures of democracy. In particular, via moment methods we derive laws for the transport, from synapse to soma, of strength, characteristic time, and dispersion. These laws lead immediately to synaptic scalings that overcome attenuation with distance. We follow this with a Neumann approximation of Green’s representation that readily produces the synaptic scaling that democratizes the peak somatic voltage response. Results are obtained for both idealized geometries and for the more realistic geometry of a rat CA1 pyramidal cell. For each measure of democratization we produce and contrast the synaptic scaling associated with treating the synapse as either a conductance change or a current injection. We find that our respective scalings agree up to a critical distance from the soma and we reveal how this critical distance decreases with decreasing branch radius.

The evolution of imperfect mimicry in hoverflies

Ideas about the evolution of imperfect mimicry are reviewed. Their relevance to the colours patterns of hoverflies (Diptera, Syrphidae) are discussed in detail. Most if not all of the hoverflies labelled as mimetic actually are mimics. The apparently poor nature of their resemblance does not prevent them from obtaining at least some protection from suitably experienced birds. Mimicry is a dominant theme of this very large family of Diptera, with at least a quarter of all species in Europe being mimetic. Hoverfly mimics fall into three major groups according to their models, involving bumblebees, honeybees and social wasps. There are striking differences in the general levels of mimetic fidelity and relative abundances of the three groups, with accurate mimicry, low abundance and polymorphism characterizing the bumblebee mimics: more than half of all the species of bumblebee mimics are polymorphic. Mimics of social wasps tend to be poor mimics, have high relative abundance, and polymorphism is completely absent. Bumblebee models fall into a small number of Muellerian mimicry rings which are very different between the Palaearctic and Nearctic regions. Social wasps and associated models form one large Muellerian complex. Together with honeybees, these complexes probably form real clusters of forms as perceived by many birds. All three groups of syrphid mimics contain both good and poor mimics; some mimics are remarkably accurate, and have close morphological and behavioural resemblance. At least some apparently 'poor' mimetic resemblances may be much closer in birds' perception than we imagine, and more work needs to be done on this. Bumblebees are the least noxious and wasps the most noxious of the three main model groups. The basis of noxiousness is different, with bumblebees being classified as non-food, whereas honeybees and wasps are nasty-tasting and (rarely) stinging. The distribution of mimicry is exactly what would be expected from this ordering, with polymorphic and accurate forms being a key feature of mimics of the least noxious models, while highly noxious models have poor-quality mimicry. Even if the high abundance of many syrphid mimics relative to their models is a recent artefact of man-made environmental change, this does not preclude these species from being mimics. It seems unlikely that bird predation actually controls the populations of adult syrphids. Being rare relative to a model may have promoted or accelerated the evolution of perfect mimicry: theoretically this might account for the pattern of rare good mimics and abundant poor ones, but the idea is intrinsically unlikely. Many mimics seem to have hour-to-hour abundances related to those of their models, presumably as a result of behavioural convergence. We need to know much more about the psychology of birds as predators. There are at least four processes that need elucidating: (a) learning about the noxiousness of models; (b) the erasing of that learning through contact with mimics (extinction, or learned forgetting); (c) forgetting; (d) deliberate risk-taking and the physiological states that promote it. Johnston's (2002) model of the stabilization of imperfect mimicry by kin selection is unlikely to account for the colour patterns of hoverflies. Sherratt's (2002) model of the influence of multiple models potentially accounts for all the patterns of hoverfly mimicry, and is the most promising avenue for testing.

Mode locking in a spatially extended neuron model: active soma and compartmental tree

Understanding the mode-locked response of excitable systems to periodic forcing has important applications in neuroscience. For example it is known that spatially extended place cells in the hippocampus are driven by the theta rhythm to generate a code conveying information about spatial location. Thus it is important to explore the role of neuronal dendrites in generating the response to periodic current injection. In this paper we pursue this using a compartmental model, with linear dynamics for each compartment, coupled to an active soma model that generates action potentials. By working with the piece-wise linear McKean model for the soma we show how the response of the whole neuron model (soma and dendrites) can be written in closed form. We exploit this to construct a stroboscopic map describing the response of the spatially extended model to periodic forcing. A linear stability analysis of this map, together with a careful treatment of the non-differentiability of the soma model, allows us to construct the Arnol'd tongue structure for 1:q states (one action potential for q cycles of forcing). Importantly we show how the presence of quasi-active membrane in the dendrites can influence the shape of tongues. Direct numerical simulations confirm our theory and further indicate that resonant dendritic membrane can enlarge the windows in parameter space for chaotic behavior. These simulations also show that the spatially extended neuron model responds differently to global as opposed to point forcing. In the former case spatio-temporal patterns of activity within an Arnol'd tongue are standing waves, whilst in the latter they are traveling waves.