Phase Response Curve Research Articles

Phase-Amplitude Response Functions for Transient-State Stimuli

The phase response curve (PRC) is a powerful tool to study the effect of a perturbation on the phase of an oscillator, assuming that all the dynamics can be explained by the phase variable. However, factors like the rate of convergence to the oscillator, strong forcing or high stimulation frequency may invalidate the above assumption and raise the question of how is the phase variation away from an attractor. The concept of isochrons turns out to be crucial to answer this question; from it, we have built up Phase Response Functions (PRF) and, in the present paper, we complete the extension of advancement functions to the transient states by defining the Amplitude Response Function (ARF) to control changes in the transversal variables. Based on the knowledge of both the PRF and the ARF, we study the case of a pulse-train stimulus, and compare the predictions given by the PRC-approach (a 1D map) to those given by the PRF-ARF-approach (a 2D map); we observe differences up to two orders of magnitude in favor of the 2D predictions, especially when the stimulation frequency is high or the strength of the stimulus is large. We also explore the role of hyperbolicity of the limit cycle as well as geometric aspects of the isochrons. Summing up, we aim at enlightening the contribution of transient effects in predicting the phase response and showing the limits of the phase reduction approach to prevent from falling into wrong predictions in synchronization problems.List of AbbreviationsPRC phase response curve, phase resetting curve.PRF phase response function.ARF amplitude response function.

Theoretical Control of the Plant Biological Clock by the Periodic Dark Pulses

The endogenous circadian rhythms close to a 24-h period are observed by nearly all living organisms including bacteria, plants, animals, and humans. Plants use the circadian clocks to coordinate gene expression, metabolism, physiology and growth. The phase of circadian rhythm can be changed (advanced or delayed) by some stimuli. We used a dark pulse in continuous light pulse (LL) in the stimuli. During cultivation of plants, the use of dark pulses in LL is more suitable than that of light pulse in DD, because the light energy for photosynthesis is sufficient in LL with short dark pulses. By application of a 2-h dark pulse at subjective dusk, a small phase advance was induced (about 2-h phase advance). In contrast, by the application of a 2-h dark pulse at subjective dawn, a phase was induced (about 4-h phase delay).With knowledge of behavior of rhythms for dark pulse, we can control circadian clock and promote productivity. We described the phase response curve (PRC) of 2-h dark pulse by a sinusoidal curve with second harmonic component. By application of the PRC, we evaluated the relationship of plant growth and the rhythm of the circadian clock control with dark pulses.

Desynchronization boost by non-uniform coordinated reset stimulation in ensembles of pulse-coupled neurons

Several brain diseases are characterized by abnormal neuronal synchronization. Desynchronization of abnormal neural synchrony is theoretically compelling because of the complex dynamical mechanisms involved. We here present a novel type of coordinated reset (CR) stimulation. CR means to deliver phase resetting stimuli at different neuronal sub-populations sequentially, i.e., at times equidistantly distributed in a stimulation cycle. This uniform timing pattern seems to be intuitive and actually applies to the neural network models used for the study of CR so far. CR resets the population to an unstable cluster state from where it passes through a desynchronized transient, eventually resynchronizing if left unperturbed. In contrast, we show that the optimal stimulation times are non-uniform. Using the model of weakly pulse-coupled neurons with phase response curves, we provide an approach that enables to determine optimal stimulation timing patterns that substantially maximize the desynchronized transient time following the application of CR stimulation. This approach includes an optimization search for clusters in a low-dimensional pulse coupled map. As a consequence, model-specific non-uniformly spaced cluster states cause considerably longer desynchronization transients. Intriguingly, such a desynchronization boost with non-uniform CR stimulation can already be achieved by only slight modifications of the uniform CR timing pattern. Our results suggest that the non-uniformness of the stimulation times can be a medically valuable parameter in the calibration procedure for CR stimulation, where the latter has successfully been used in clinical and pre-clinical studies for the treatment of Parkinson's disease and tinnitus.

Neural dynamics and information representation in microcircuits of motor cortex

The brain has to analyze and respond to external events that can change rapidly from time to time, suggesting that information processing by the brain may be essentially dynamic rather than static. The dynamical features of neural computation are of significant importance in motor cortex that governs the process of movement generation and learning. In this paper, we discuss these features based primarily on our recent findings on neural dynamics and information coding in the microcircuit of rat motor cortex. In fact, cortical neurons show a variety of dynamical behavior from rhythmic activity in various frequency bands to highly irregular spike firing. Of particular interest are the similarity and dissimilarity of the neuronal response properties in different layers of motor cortex. By conducting electrophysiological recordings in slice preparation, we report the phase response curves (PRCs) of neurons in different cortical layers to demonstrate their layer-dependent synchronization properties. We then study how motor cortex recruits task-related neurons in different layers for voluntary arm movements by simultaneous juxtacellular and multiunit recordings from behaving rats. The results suggest an interesting difference in the spectrum of functional activity between the superficial and deep layers. Furthermore, the task-related activities recorded from various layers exhibited power law distributions of inter-spike intervals (ISIs), in contrast to a general belief that ISIs obey Poisson or Gamma distributions in cortical neurons. We present a theoretical argument that this power law of in vivo neurons may represent the maximization of the entropy of firing rate with limited energy consumption of spike generation. Though further studies are required to fully clarify the functional implications of this coding principle, it may shed new light on information representations by neurons and circuits in motor cortex.

Impact of neuronal heterogeneity on correlated colored noise-induced synchronization

Synchronization plays an important role in neural signal processing and transmission. Many hypotheses have been proposed to explain the origin of neural synchronization. In recent years, correlated noise-induced synchronization has received support from many theoretical and experimental studies. However, many of these prior studies have assumed that neurons have identical biophysical properties and that their inputs are well modeled by white noise. In this context, we use colored noise to induce synchronization between oscillators with heterogeneity in both phase-response curves and frequencies. In the low noise limit, we derive novel analytical theory showing that the time constant of colored noise influences correlated noise-induced synchronization and that oscillator heterogeneity can limit synchronization. Surprisingly, however, heterogeneous oscillators may synchronize better than homogeneous oscillators given low input correlations. We also find resonance of oscillator synchronization to colored noise inputs when firing frequencies diverge. Collectively, these results prove robust for both relatively high noise regimes and when applied to biophysically realistic spiking neuron models, and further match experimental recordings from acute brain slices.

Impact of Photoperiod on the Circadian Cellulose and Cellulase Rhythms in the Digestive System of Silkworm, Bombyx Mori

Circadian cellulose and cellulase rhythms were studied in the digestive system of Bombyx mori under 12 hr light-dark cycle (LD), continuous light (LL) and continuous dark (DD) conditions. Since, the symbiotic microbes are associated with cellulose digestion in the silkworm; both the rhythms are presumed to have microbial origin. The rhythms were analyzed in terms of phase response curves and interpreted as cellulose synthetic cycles (CS cycles) and cellulase enzyme synthetic cycles (CES cycles) in gut wall and cellulose uptake cycles (CU cycles) and cellulase enzyme release cycles (CER cycles) in gut lumen. The cellulose rhythm in the gut wall maintained 7 CS cycles each with a duration of 3.4 hr and the 24 hr free running time remained unchanged under altered photoperiodic conditions. In gut lumen, the rhythm showed 7 CU cycles, each with a duration of 3.4 hr under LD and 6 CU cycles of 4.0 hr each under LL and DD, with the result, the 24 hr free running period of the rhythm was clock shifted to 28 hr both under LL and DD. The cellulase rhythm in the gut wall maintained 7 CES cycles with a duration of 3.4 hr each under LD and DD and 9 cycles each with a duration of 2.7 hr under LL. The 24 hr free running time of the cellulase rhythm of LD and DD, was thus advanced to 19 hr under LL. The cellulase in gut lumen maintained 6 CER cycles with a duration of 4.0 hr each, under LD, LL and DD and hence the 24 hr free running time of the rhythm was not affected. Further analysis of data showed that cellulose synthesis was stimulated by light cues, while its uptake from gut lumen through enhanced microbial cellulase activity was favoured by both diet and light cues.

Dynamical Systems Guided Design and Analysis of Silicon Oscillators for Central Pattern Generators

In this paper, a dynamical systems (DS) approach is proposed for the analysis and design of bio-inspired silicon central pattern generator (CPG) systems. Based on this approach, a new leaky-integrate-and-leaky-discharge oscillator circuit is proposed that has dynamical properties closer to biological half-center oscillators while being power and area efficient. The membrane potential charges and discharges through a single resistor eliminating mismatch in charging and discharging phases. Switched-capacitor (SC) and floating-gate wide input linear range operational transconductance amplifier (FGOTA) based approaches have been proposed to implement the resistor. Both approaches enable controllable and large resistances in a small area. Oscillation frequency can be easily controlled by the frequency of switching in SC based and bias current in FGOTA based implementations, which are very useful for global change of oscillation frequency in an array of oscillators. Dynamical systems analysis has shown that when it is used as a single oscillator, the proposed circuit is able to produce a phase response curve (PRC) close to that of a lamprey CPG system. By applying averaging theory to a system of coupled oscillators, the averaged <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">G</i> functions for unidirectional and bidirectional coupling cases are obtained. Analysis of these functions shows our circuit's superior capability to achieve entrainment when driven by a periodic input (e.g., from sensory feedback) and reach equilibrium even with high frequency mismatch.

Higher-Order Spike Triggered Analysis of Neural Oscillators

For the purpose of elucidating the neural coding process based on the neural excitability mechanism, researchers have recently investigated the relationship between neural dynamics and the spike triggered stimulus ensemble (STE). Ermentrout et al. analytically derived the relational equation between the phase response curve (PRC) and the spike triggered average (STA). The STA is the first cumulant of the STE. However, in order to understand the neural function as the encoder more explicitly, it is necessary to elucidate the relationship between the PRC and higher-order cumulants of the STE. In this paper, we give a general formulation to relate the PRC and the nth moment of the STE. By using this formulation, we derive a relational equation between the PRC and the spike triggered covariance (STC), which is the covariance of the STE. We show the effectiveness of the relational equation through numerical simulations and use the equation to identify the feature space of the rat hippocampal CA1 pyramidal neurons from their PRCs. Our result suggests that the hippocampal CA1 pyramidal neurons oscillating in the theta frequency range are commonly sensitive to inputs composed of theta and gamma frequency components.

Human phase response curve to a single 6.5 h pulse of short‐wavelength light

The photic resetting response of the human circadian pacemaker depends on the timing of exposure, and the direction and magnitude of the resulting shift is described by a phase response curve (PRC). Previous PRCs in humans have utilized high-intensity polychromatic white light. Given that the circadian photoreception system is maximally sensitive to short-wavelength visible light, the aim of the current study was to construct a PRC to blue (480 nm) light and compare it to a 10,000 lux white light PRC constructed previously using a similar protocol. Eighteen young healthy participants (18-30 years) were studied for 9-10 days in a time-free environment. The protocol included three baseline days followed by a constant routine (CR) to assess initial circadian phase. Following this CR, participants were exposed to a 6.5 h 480 nm light exposure (11.8 μW cm(-2), 11.2 lux) following mydriasis via a modified Ganzfeld dome. A second CR was conducted following the light exposure to re-assess circadian phase. Phase shifts were calculated from the difference in dim light melatonin onset (DLMO) between CRs. Exposure to 6.5 h of 480 nm light resets the circadian pacemaker according to a conventional type 1 PRC with fitted maximum delays and advances of -2.6 h and 1.3 h, respectively. The 480 nm PRC induced ∼75% of the response of the 10,000 lux white light PRC. These results may contribute to a re-evaluation of dosing guidelines for clinical light therapy and the use of light as a fatigue countermeasure.

Cardiac Atrial Circadian Rhythms in PERIOD2::LUCIFERASE and per1:luc Mice: Amplitude and Phase Responses to Glucocorticoid Signaling and Medium Treatment

Circadian rhythms in cardiac function are apparent in e.g., blood pressure, heart rate, and acute adverse cardiac events. A circadian clock in heart tissue has been identified, but entrainment pathways of this clock are still unclear. We cultured tissues of mice carrying bioluminescence reporters of the core clock genes, period 1 or 2 (per1luc or PER2LUC) and compared in vitro responses of atrium to treatment with medium and a synthetic glucocorticoid (dexamethasone [DEX]) to that of the suprachiasmatic nucleus (SCN) and liver. We observed that PER2LUC, but not per1luc is rhythmic in atrial tissue, while both per1luc and PER2LUC exhibit rhythmicity in other cultured tissues. In contrast to the SCN and liver, both per1luc and PER2LUC bioluminescence amplitudes were increased in response to DEX treatment, and the PER2LUC amplitude response was dependent on the time of treatment. Large phase-shift responses to both medium and DEX treatments were observed in the atrium, and phase responses to medium treatment were not attributed to serum content but the treatment procedure itself. The phase-response curves of atrium to both DEX and medium treatments were found to be different to the liver. Moreover, the time of day of the culturing procedure itself influenced the phase of the circadian clock in each of the cultured tissues, but the magnitude of this response was uniquely large in atrial tissue. The current data describe novel entrainment signals for the atrial circadian clock and specifically highlight entrainment by mechanical treatment, an intriguing observation considering the mechanical nature of cardiac tissue.

Chronotype differences in Drosophila are enhanced by semi-natural conditions

Morning and evening chronotypes of sleep/wake cycles in humans are often metaphorically termed as 'larks' and 'owls'. We derived Drosophila populations early and late, displaying lark- and owl-like emergence chronotypes by imposing selection for adult emergence during morning and evening hours. Preference for morning and evening emergence in these populations was accompanied by divergence in their circadian period (τ) and photic phase response curve. To test if lark- and owl-like emergence chronotypes displayed by these flies under weak environmental cycles of the laboratory would also persist in nature where several zeitgebers are present in the strongest form, we examined the emergence rhythm of early and late flies under semi-natural conditions. The early and late flies not only continued to exhibit divergent emergence waveforms under semi-natural conditions, the differences became even more prominent. However, phases of early and late emergence waveforms did not match natural morning and evening transitions, unlike that observed under laboratory conditions. These results thus provide evidence consistent with the notion that chronotypes are the result of interactions between circadian clocks and natural environmental cycles.

Distinct Phase Relationships between Suprachiasmatic Molecular Rhythms, Cerebral Cortex Molecular Rhythms, and Behavioral Rhythms in Early Runner (CAST/EiJ) and Nocturnal (C57BL/6J) Mice

We have previously established that CAST/EiJ (CAST) mice differ from normal mice, such as C57BL/6J (B6), in the timing of wheel-running onset relative to light/dark cycles. These mice provide an animal model for studies of the genetic and neurobiological basis for circadian phase misalignment in humans. Neither differences in endogenous circadian period nor the shape of the photic phase response curve explain the difference in the timing of activity onset between CAST and B6 mice, suggesting a mechanism downstream of the circadian clock. Here, we further test the hypothesis that the two strains differ with respect to circadian oscillations at the molecular level. Sleep/wake cycles were examined and rhythms of Period1 (Per1) and Period2 (Per2) expression were measured in the cerebral cortex, suprachiasmatic nucleus (SCN), and other hypothalamic regions. Basic sleep and molecular research laboratory. Male mice of the B6 and CAST inbred strains. None. Sleep/wake cycles were advanced by approximately 4 h in CAST mice relative to B6 mice. This was paralleled by phase-advanced rhythms of Per1 and Per2 expression, as measured byin situ hybridization, in the cerebral cortex of CAST relative to B6. By contrast, the timing of circadian oscillations and the photic induction ofPer1 and Per2 expression in the SCN were unaffected by strain. The advanced phase of wheel running and sleep/wake cycles in CAST mice relative to B6 mice is apparently not associated with differences in molecular oscillations in the SCN clock itself, but most likely in mechanisms downstream of the SCN clock. CAST mice may therefore provide a model system to investigate circadian downstream mechanisms underlying unusual patterns of entrainment to the ambient photoperiod. Jiang P; Franklin KM; Duncan MJ; O'Hara BF; Wisor JP. Distinct phase relationships between suprachiasmatic molecular rhythms, cerebral cortex molecular rhythms, and behavioral rhythms in early runner (CAST/EiJ) and nocturnal (C57BL/6J) mice. SLEEP 2012;35(10):1385-1394.

Probabilistic convergence guarantees for type-II pulse-coupled oscillators

We show that a large class of pulse-coupled oscillators converge with high probability from random initial conditions on a large class of graphs with time delays. Our analysis combines previous local convergence results, probabilistic network analysis, and a classification scheme for type-II phase response curves to produce rigorous lower bounds for convergence probabilities based on network density. These results suggest methods for the analysis of pulse-coupled oscillators, and provide insights into the balance of excitation and inhibition in the operation of biological type-II phase response curves and also the design of decentralized and minimal clock synchronization schemes in sensor nets.

The Input Signal Step Function (ISSF), a Standard Method to Encode Input Signals in SBML Models with Software Support, Applied to Circadian Clock Models

Time-dependent light input is an important feature of computational models of the circadian clock. However, publicly available models encoded in standard representations such as the Systems Biology Markup Language (SBML) either do not encode this input or use different mechanisms to do so, which hinders reproducibility of published results as well as model reuse. The authors describe here a numerically continuous function suitable for use in SBML for models of circadian rhythms forced by periodic light-dark cycles. The Input Signal Step Function (ISSF) is broadly applicable to encoding experimental manipulations, such as drug treatments, temperature changes, or inducible transgene expression, which may be transient, periodic, or mixed. It is highly configurable and is able to reproduce a wide range of waveforms. The authors have implemented this function in SBML and demonstrated its ability to modify the behavior of publicly available models to accurately reproduce published results. The implementation of ISSF allows standard simulation software to reproduce specialized circadian protocols, such as the phase-response curve. To facilitate the reuse of this function in public models, the authors have developed software to configure its behavior without any specialist knowledge of SBML. A community-standard approach to represent the inputs that entrain circadian clock models could particularly facilitate research in chronobiology.

Human phase response curve to intermittent blue light using a commercially available device

Light shifts the timing of the circadian clock according to a phase response curve (PRC). To date, all human light PRCs have been to long durations of bright white light. However, melanopsin, the primary photopigment for the circadian system, is most sensitive to short wavelength blue light. Therefore, to optimise light treatment it is important to generate a blue light PRC.We used a small, commercially available blue LED light box, screen size 11.2 × 6.6 cm at ∼50 cm, ∼200 μW cm(−2), ∼185 lux. Subjects participated in two 5 day laboratory sessions 1 week apart. Each session consisted of circadian phase assessments to obtain melatonin profiles before and after 3 days of free-running through an ultradian light–dark cycle (2.5 h wake in dim light, 1.5 h sleep in the dark), forced desynchrony protocol. During one session subjects received intermittent blue light (three 30 min pulses over 2 h) once a day for the 3 days of free-running, and in the other session (control) they remained in dim room light, counterbalanced. The time of blue light was varied among subjects to cover the entire 24 h day. For each individual, the phase shift to blue light was corrected for the free-run determined during the control session. The blue light PRC had a broad advance region starting in the morning and extending through the afternoon. The delay region started a few hours before bedtime and extended through the night. This is the first PRC to be constructed to blue light and to a stimulus that could be used in the real world.

Intrinsic heterogeneity in oscillatory dynamics limits correlation-induced neural synchronization

Synchronous neural oscillations are found throughout the brain and are thought to contribute to neural coding and the propagation of activity. Several proposed mechanisms of synchronization have gained support through combined theoretical and experimental investigation, including mechanisms based on coupling and correlated input. Here, we ask how correlation-induced synchrony is affected by physiological heterogeneity across neurons. To address this question, we examined cell-to-cell differences in phase-response curves (PRCs), which characterize the response of periodically firing neurons to weak perturbations. Using acute slice electrophysiology, we measured PRCs across a single class of principal neurons capable of sensory-evoked oscillations in vivo: the olfactory bulb mitral cells (MCs). Periodically firing MCs displayed a broad range of PRCs, each of which was well fit by a simple three-parameter model. MCs also displayed differences in firing rate-current relationships and in preferred firing rate ranges. Both the observed PRC heterogeneity and moderate firing rate differences (∼10 Hz) separately reduced the maximum correlation-induced synchrony between MCs by up to 25-30%. Simulations further demonstrated that these components of heterogeneity alone were sufficient to account for the difference in synchronization among heterogeneous vs. homogeneous populations in vitro. Within this simulation framework, independent modulation of specific PRC features additionally revealed which aspects of PRC heterogeneity most strongly impact correlation-induced synchronization. Finally, we demonstrated good agreement of novel mathematical theory with our experimental and simulation results, providing a theoretical basis for the influence of heterogeneity on correlation-induced neural synchronization.

Adaptive Temperature Compensation in Circadian Oscillations

A temperature independent period and temperature entrainment are two defining features of circadian oscillators. A default model of distributed temperature compensation satisfies these basic facts yet is not easily reconciled with other properties of circadian clocks, such as many mutants with altered but temperature compensated periods. The default model also suggests that the shape of the circadian limit cycle and the associated phase response curves (PRC) will vary since the average concentrations of clock proteins change with temperature. We propose an alternative class of models where the twin properties of a fixed period and entrainment are structural and arise from an underlying adaptive system that buffers temperature changes. These models are distinguished by a PRC whose shape is temperature independent and orbits whose extrema are temperature independent. They are readily evolved by local, hill climbing, optimization of gene networks for a common quality measure of biological clocks, phase anticipation. Interestingly a standard realization of the Goodwin model for temperature compensation displays properties of adaptive rather than distributed temperature compensation.

Optimal entrainment of neural oscillator ensembles

In this paper, we derive the minimum-energy periodic control that entrains an ensemble of structurally similar neural oscillators to a desired frequency. The state-space representation of a nominal oscillator is reduced to a phase model by computing its limit cycle and phase response curve, from which the optimal control is derived by using formal averaging and the calculus of variations. We focus on the case of a 1:1 entrainment ratio and suggest a simple numerical method for approximating the optimal controls. The method is applied to asymptotically control the spiking frequency of neural oscillators modeled using the Hodgkin–Huxley equations. Simulations are used to illustrate the optimality of entrainment controls derived using phase models when applied to the original state-space system, which is crucial for using phase models in control synthesis for practical applications. This work addresses a fundamental problem in the field of neural dynamics and provides a theoretical contribution to the optimal frequency control of uncertain oscillating systems.

Phase response curves of subthalamic neurons measured with synaptic input and current injection

Infinitesimal phase response curves (iPRCs) provide a simple description of the response of repetitively firing neurons and may be used to predict responses to any pattern of synaptic input. Their simplicity makes them useful for understanding the dynamics of neurons when certain conditions are met. For example, the sizes of evoked phase shifts should scale linearly with stimulus strength, and the form of the iPRC should remain relatively constant as firing rate varies. We measured the PRCs of rat subthalamic neurons in brain slices using corticosubthalamic excitatory postsynaptic potentials (EPSPs; mediated by both AMPA- and NMDA-type receptors) and injected current pulses and used them to calculate the iPRC. These were relatively insensitive to both the size of the stimulus and the cell's firing rate, suggesting that the iPRC can predict the response of subthalamic nucleus cells to extrinsic inputs. However, the iPRC calculated using EPSPs differed from that obtained using current pulses. EPSPs (normalized for charge) were much more effective at altering the phase of subthalamic neurons than current pulses. The difference was not attributable to the extended time course of NMDA receptor-mediated currents, being unaffected by blockade of NMDA receptors. The iPRC provides a good description of subthalamic neurons' response to input, but iPRCs are best estimated using synaptic inputs rather than somatic current injection.

Biophysical basis of the phase response curve of subthalamic neurons with generalization to other cell types

Experimental evidence indicates that the response of subthalamic neurons to excitatory postsynaptic potentials (EPSPs) is well described by their infinitesimal phase response curves (iPRC). However, the factors controlling the shape of that iPRC, and hence controlling the way subthalamic neurons respond to synaptic input, are unclear. We developed a biophysical model of subthalamic neurons to aid in the understanding of their iPRCs; this model exhibited an iPRC type common to many subthalamic cells. We devised a method for deriving its iPRC from its biophysical properties that clarifies how these different properties interact to shape the iPRC. This method revealed why the response of subthalamic neurons is well approximated by their iPRCs and how that approximation becomes less accurate under strong fluctuating input currents. It also connected iPRC structure to aspects of cellular physiology that could be estimated in simple current-clamp experiments. This allowed us to directly compare the iPRC predicted by our theory with the iPRC estimated from the response to EPSPs or current pulses in individual cells. We found that theoretically predicted iPRCs agreed well with estimates derived from synaptic stimuli, but not with those estimated from the response to somatic current injection. The difference between synaptic currents and those applied experimentally at the soma may arise from differences in the dynamics of charge redistribution on the dendrites and axon. Ultimately, our approach allowed us to identify novel ways in which voltage-dependent conductances interact with AHP conductances to influence synaptic integration that will apply to a wide range of cell types.