Research projects of Alla Borisyuk
June 2012
My research focuses on mathematical analysis of neuronal models. It is of particular interest to me how dynamics of
neural activity are shaped by low-level processes such as single cell properties, network architecture, plasticity of
connections between cells, etc. Currently, my main focus is on questions of temporal processing in the auditory system,
and on properties of stochastic neuronal models.
Recent projects (completed in the last 2 years)
(up-to-date versions and status of preprints are at my website under “Research”: www.math.utah.edu/~borisyuk/)
Noise in synaptic conductances enables reconstruction of three stimulus-related inputs. With S. Odom
In studies of sensory processing, it is often a challenge to find out where a particular computation is performed. One of the
strategies is to perform in vivo recordings at one location, and use them to compute afferent inputs to see whether a
particular computation is performed de novo at the given processing stage or inherited from lower level structures.
The existing model-based methods focus on resolving two synaptic conductances corresponding to two distinct reversal
potentials. We present an approach enabling the reconstruction of three input conductances. Our method is based on
exploiting the stochastic nature of synaptic conductances and membrane voltage. We generalize the model to a stochastic
differential equation and derive equations for first and second moments that can be solved to find conductances. We
successfully apply the new reconstruction method to simulated data. We also explore the robustness of the method as the
assumptions of the underlying model are relaxed. We vary the noise levels, the reversal potentials, the number of stimulus
repetitions, and the accuracy of conductance variance estimation to quantify the robustness of reconstruction. These
studies pave way for the application of the method to experimental data. (Preprint is on the website)

S.E.Odom, A.Borisyuk. Estimating three synaptic conductances in a stochastic neural model. DOI:
10.1007/s10827-012-0382-z. J. Comp. Neurosci. 2012.
Separating frequencies in excitatory-inhibitory networks. With J. Best and D. Terman
We consider a situation in which individual features of the input are represented in the neural system by different
frequencies of periodic firings. Thus, if two of the features are presented concurrently, the input to the system will consist
of a superposition of two periodic trains. In this paper we present an algorithm that is capable of extracting the individual
features (individual periodic trains) from the superimposed signal (frequency separation).
We show that the algorithm can be implemented in a biophysically based excitatory-inhibitory network model. The
frequency separation process works over a range of frequencies determined by time constants of the model intrinsic
variables. It does not rely on a ``resonance'' phenomenon and is not tuned to a discrete set of frequencies. The frequency
separation is still reliable when incoming pulses are jittered. (Preprint is on the website)

Borisyuk, J. Best, D. Terman. Frequency separation by an excitatory-inhibitory network.Resubmitted after
revision. (Response from the Action Editor to the original submission: “Both reviewers are in favor of publication
after suitable revisions”.)
Spike Phase Locking in CA1 Pyramidal Neurons. With T. Broicher, P. Malerba, A. Dorval, F. Fernandez, J.White
In a project with John White, Chuck Dorval (Bioengineering, U. of Utah) and postdoctoral fellows Paola Malerba,
Tilman Broicher, and Fernando Fernandez we are working on establishing phase-locking properties of pyramidal cells in
rat hippocampal brain slices in response to (conductance-based or current-based) noisy periodic inputs. My colleagues
performed dynamic-clamp experiments to determine action potential phase-locking profiles with respect to background
conductance, average firing rate, and frequency of the sinusoidal component. Then we designed mathematical models to
explain experimental findings. We suggested that spike-rate adaptation and frequency resonance in the spike-generating
mechanism are crucial in shaping the different phase-locking profiles. Our results demonstrate that CA1 pyramidal cells
can actively change their synchronization properties in response to global changes in activity associated with different
behavioral states.
Research projects of Alla Borisyuk

June 2012
T. Broicher, P. Malerba, A. Dorval, A. Borisyuk, F. Fernandez, and J.White. Spike Phase Locking in CA1
Pyramidal Neurons depends on Background Conductance and Firing Rate. Resubmitted after revision.
(Response from the Senior Editor to the original submission: “I am pleased to inform you that your paper, <title>
has been judged potentially suitable for publication in The Journal of Neuroscience, provided appropriate
revisions are made.”)
A spectral approach to stochastic circle maps. With F. Rassoul-Agha.
In an effort to develop a mathematical theory applicable to the above biologically-motivated project on phase-locking and
related problems, we are developing, in collaboration with Firas Rassoul-Agha (Mathematics, U. of Utah) a spectral
approach to stochastic circle maps. A stochastic circle map is defined as a Markov chain on the circle. This class of
objects includes a wide range of models for firing times of periodically forced noisy neuronal models. The main tool is
spectral analysis of the transition operator of the Markov chain. We define and analyze path-wise dynamic properties of
the Markov chain, such as stochastic periodicity (or phase locking) and stochastic quasiperiodicity, and show how these
properties are read off of the transition operator spectrum geometry. We observe and explain spectral spirals in the
quasiperiodic regime, “zipping" of the spectrum, and emergence and dominance of various phase-locked states. (Preprint
is on the website)

Borisyuk, F. Rassoul-Agha. Quasiperiodicity and Phase Locking in Stochastic Circle Maps: a Spectral Approach.
Submitted.
Current projects
Glomerular processing of olfactory inputs. With R. Carey, E. Sherwood, M. Wachowiak
With Matt Wachowiak (Dept pf Physiology, U. Of Utah), his PhD student Ryan Carey and my postdoc Erik Sherwood we
have investigated an important question in olfactory coding: the degree to which mitral cell (MC) excitation patterns can
be shaped temporally by various intraglomerular processes. Individual components of the intraglomerular circuitry –
including depression at the ORN synapses the ET cell, feedforward and feedback inhibition, and the MC cell itself – were
implemented in a hierarchy of computational models and evaluated based on their ability to transform realistic ORN
inputs to well-patterned MC outputs, even at high sniff frequencies. This study represents the first to simulate glomerular
responses to real ORN input signals, using recently-published results to constrain the possible pathways. Our results
indicate that ET cells may serve a critical functional role in determining the temporal dynamics of MC responses, and
ensuring the maintenance of strongly sniff-modulated MC firing patterns at higher sniff frequencies. In these models,
feedforward and recurrent inhibition seem to modulate peak firing rates, but do not contribute substantially to the finescale temporal structure of MC responses to each sniff.

R. Carey, E. Sherwood, A. Borisyuk M. Wachowiak. “Glomerular processing of olfactory inputs”, preprint
Neuronal transmission of timing precision: dependence on intrinsic and synaptic properties. With H. Brooks
Precision of spike timing and its role in information processing is one of the key themes in systems neuroscience. As
signals travel through nervous systems, the precision and reliability of the timing information is altered. Interestingly,
depending on the particular system, the timing precision can be improved or reduced through propagation. This raises the
issue of what are the properties that allow for the improvement or deterioration of the timing information? Motivated by
processing of timing information in the auditory system, with undergraduate student Heather Brooks we are looking at the
cellular and synaptic properties that allow improving or decreasing the precision of spike timing through synaptic
transmission. We have shown in minimal neuronal models (ingrate-and-fire and Morris-Lecar) that both improvement and
deterioration of spike-time precision is possible, depending on the input times distribution and synaptic strength. The
Research projects of Alla Borisyuk
June 2012
work is ongoing to include more nuanced intrinsic and synaptic properties. (Results will be presented at CNS 2012 and
SIAM Life Sciences conferences this summer; manuscript in preparation)
Mechanisms of slow-rise-time selectivity in anuran auditory midbrain. With S. Odom and G. Rose
Experiments in frog Hyla versicolor have shown behavioral and cellular selectivity to slow rise time of amplitude
modulation of auditory stimuli. In collaboration with Gary Rose (Biology, U. of Utah) and my Ph.D. student Steve Odom
we are working on establishing the mechanisms of this selectivity. We proposed three different mechanisms, two of which
have not been previously discussed in literature. First, involving fast-rise-sensitive inhibition at earlier processing levels;
second, involving steady state inhibition slowly inactivated by the stimuli; and third, based on previously established
interval counting properties of cells. For each of these mechanisms we demonstrate compatibility with existing data
(including data obtained with conductance-reconstruction methods described above) and make predictions that will allow
distinguishing between different mechanisms in future experiments. (Presentation and a poster are on the website, the
manuscript is in preparation).
Some phase transitions in very sparsely connected random neuronal networks. With S. Ahn and W. Just.
In my earlier work (Ahn et al., 2010), we considered spread of activity over random digraphs, with refractoriness-enabled
elements. Our numerical simulations suggested that there are phase transitions in probabilities of various outcomes of the
dynamics, as connection probability is varied. In present work we explore phase transitions in much more detail, and also
derive analytical estimates for transition parameters. (Manuscript is in preparation)
Coding interaural time differences with cochlear implants. With C. Hokama.
In this project, with Chad Hokama (M.S. student from Computational and Engineering Science program) we study the
responses of cells in the inferior colliculus (a crucial auditory node) to the interaural time differences (ITDs) in the
amplitude modulated signals. We are especially interested in responses to electrically generated inputs, as in cochlear
implants. Existing, highly simplified, model helps explain many of the experimental findings. However, the simplicity of
the model precludes it from addressing other response features, notably those that depend on stochasticity in the system.
In our work, we have combined this model with recent powerful model of electrically stimulated auditory nerve. With the
new combined system we are now working to explain the shifts in the best ITD with changes in intensity and in the
interaural level difference, and the variability of responses. (This work is ongoing)
Learning in a biophysical network: constraints set by population oscillations
Experiments in insect antennal lobe (AL; analogue of olfactory bulb) show that population activity in a response to a
conditioned stimulus exhibits a strong oscillatory component, whereas individual cells are not oscillators. We propose to
use as a model of the AL an idealized excitatory-inhibitory network, generating population oscillations from nonoscillatory noisy elements. Tuning the network into the right frequency regime suggests constraints on the network
connectivity, in particular it suggests that E-E connections are required. Qualitatively matching the observed learninginduced changes in oscillation properties, suggests how the network parameters are affected by learning, namely that the
strength of excitation and inhibition are both increased in such a way as to retain the same excitation-inhibition ratio.
(Presentation is on the website)
Role of dendrites in noise-induced synchronization. With P. Bressloff
Many types of epilepsy have been traced to mutations in somatic and dendritic ion channels. At the same time, seizures
have long been associated with synchronization in networks of cells. In this project we are investigating how changes in
the dendrite affect tendency of the cells towards synchronization. We focus on synchronization of uncoupled neurons
driven to synchrony by a common noisy input; as may occur when neighboring tissue is recruited by the seizure focus. We
use Lyapunov exponents (introduced in this context by Teramae and Tanaka) as a measure for noise-induced
synchronization. We extend the theory to include dendrites via two different approaches: first, treating the soma and the
dendrite as a single oscillator described by the dendritic phase-resetting-curve; and second, treating the somatic oscillator
Research projects of Alla Borisyuk
June 2012
as receiving input filtered by the dendrite. We demonstrate that either approach can be used in the case of passive
dendrites and some of the active currents, including non-uniform spatial channel distribution. We find that some of the
epilepsy-implicated currents can have either synchronizing or de-synchronizing effect depending on the channels’ location
along the dendrite and that distal dendrites can have a stronger synchronizing effect than proximal ones if the “synaptic
democracy” is included in the model. (Presentation is on the website)
Earlier Projects (Reprints are available on my website)
Dynamic clustering: reduction to a discrete model.
S. Ahn, B.H. Smith, A. Borisyuk, D. Terman. (2010) Analyzing Neuronal Networks Using Discrete-Time Dynamics, Physica D:
Nonlinear phenomena 239(9): 515-528
Noise-driven rhythmogenesis
W.H. Nesse, A. Borisyuk, P.C. Bressloff. (2008) Fluctuation-Driven rhythmogenesis in an excitatory network with slow adaptation, J
Comput Neurosci. 25(2): 317-33
Dynamic regimes in a synaptically coupled network
J. Best, A. Borisyuk, J. Rubin, D. Terman, M. Wechselberger. (2005) The dynamic range of bursting in a network of synaptically
coupled square-wave bursting respiratory pacemaker cells, SIAM J. of Appl. Dyn. Syst. 4: 1107-1139
Activation patterns in honeybee antennal lobe: correlates of behavior.
A. Borisyuk, B. H. Smith. (2004) Odor interactions and learning in a model of the insect antennal lobe. Neurocomputing 58-60: 10411047
Sound localization
A. Borisyuk, M. N. Semple, J. Rinzel. Adaptation and inhibition underlie responses to time-varying interaural phase cues in a model
of inferior colliculus neurons. J. Neurophysiol. 88: 2134-2146, 2002
A. Borisyuk, M. N. Semple, J. Rinzel.Computational model for the dynamic aspects of sound processing in the auditory midbrain.
Neurocomputing 38: 1127-1134, 2001