On a response characteristic of a mathematical neuron model

A mathematical neuron model in the form of a nonlinear difference equation is proposed and its response characteristic is investigated.If a sequence of pulses with a fixed frequency is applied to the neuron model as an input, and the amplitude of the input pulses is progressively decreased, the firing frequency of the neuron model, regarded as the output, also decreases. The relationship between them is quite complicated, but a mathematical investigation reveals that it takes the form of an extended Cantor's function. This result explains the unusual and unsuspected phenomenon which was found by L. D. Harmon in experimental studies with his transistor neuron models.Besides this, as an analogue of our mathematical neuron model, a very simple circuit composed of a delay line and a negative resistance element is presented and discussed.     

Mathematical properties of responses of a neuron model

Recently, Nagumo and Sato proposed a mathematical neuron model in the form of a nonlinear difference equation and investigated its response characteristic. The result showed that the input-output relationship of the neuron model is quite complicated and takes the form of an extended Cantor's function. It also explained the unusual and unsuspected phenomenon found by Harmon in experimental studies with his transistor neuron model. — In this paper, a fraction representation of a sequence of pulses is proposed. A mathematical treatment of the same neuron model based on the representation gives the same result as in the previous paper. Moreover, many mathematical properties, including the one where the ratio of the number of 1's contained in a cycle of a sequence to the length of the cycle gives any rational number between 0 and 1, were obtained by investigating sequences generated by the model.     

Coupling-induced population synchronization in an excitatory population of subthreshold Izhikevich neurons

We consider an excitatory population of subthreshold Izhikevich neurons which exhibit noise-induced firings. By varying the coupling strength J, we investigate population synchronization between the noise-induced firings which may be used for efficient cognitive processing such as sensory perception, multisensory binding, selective attention, and memory formation. As J is increased, rich types of population synchronization (e.g., spike, burst, and fast spike synchronization) are found to occur. Transitions between population synchronization and incoherence are well described in terms of an order parameter $\mathcal{O}$ . As a final step, the coupling induces oscillator death (quenching of noise-induced spikings) because each neuron is attracted to a noisy equilibrium state. The oscillator death leads to a transition from firing to non-firing states at the population level, which may be well described in terms of the time-averaged population spike rate $\overline{R}$ . In addition to the statistical-mechanical analysis using $\mathcal{O}$ and $\overline{R}$ , each population and individual state are also characterized by using the techniques of nonlinear dynamics such as the raster plot of neural spikes, the time series of the membrane potential, and the phase portrait. We note that population synchronization of noise-induced firings may lead to emergence of synchronous brain rhythms in a noisy environment, associated with diverse cognitive functions.     

Evidence that the intrinsic membrance protein LHCII in thylakoids is necessary for maintaining localized\Delta \tilde \mu _{H^ + } energy coupling

This work tested the hypothesis that thylakoid localized proton-binding domains, suggested to be involved in localized -driven ATP formation, are maintained with the involvement of several membrane proteins, including the LHCII (Laszlo, J. A., Baker, G. M., and Dilley, R. A. (1984) Biochim. Biophys. Acta 764, 160–169), which comprises about 50% of the total thylakoid protein. The concept we have in mind is that several membrane proteins cooperate to shield a localized proton diffusion pathway from direct contact with the lumen, thus providing a physical barrier to H⁺ equilibration between the sequestered domains and the lumen. A barely mutant,chlorina f ₂, that lacks Chl b and does not accumulate some of the LHCII proteins, was tested for its capacity to carry out localized-proton gradient-dependent ATP formation. Two previously developed assays permit clear discrimination between localized and delocalized gradient-driven ATP formation. Those assays include the effect of a permeable buffer, pyridine, on the number of single-turnover flashes needed to reach the energetic threshold for ATP formation and the more recently developed assay for lumen pH using 8-hydroxy-1,3,6-pyrene trisulfonic acid as a lumenally loaded pH-sensitive fluorescent probe. By those two criteria, the wild-type barley thylakoids revealed either a localized or a delocalized energy coupling mode under low- or high-salt storage conditions, respectively. Addition of Ca⁺⁺ to the high-salt storage medium caused those thylakoids to maintain a localized energy-coupling response, as previously observed for pea thylakoids. In contrast, thechlorina f ₂ mutant thylakoids had an active delocalized energy coupling activity but did not show localized energy coupling under any conditions, and added Ca⁺⁺ to the thylakoid storage medium did not alter the delocalized energy coupling mode. One interpretation of the results is that the absence of the LHCII polypeptides produces a leaky pathway for protons which allows the gradient to equilibrate with the lumen under all conditions. Another interpretation is possible but seems less likely, that being that the absence of the LHCII polypeptides in some way causes the proposed Ca⁺⁺ -gated H⁺ flux site on the membrane sector (CF₀) of the energy coupling complex to lose its gating function.     

A second-generation computational modeling of cardiac electrophysiology: response of action potential to ionic concentration changes and metabolic inhibition

Background

Cardiac arrhythmias are becoming one of the major health care problem in the world, causing numerous serious disease conditions including stroke and sudden cardiac death. Furthermore, cardiac arrhythmias are intimately related to the signaling ability of cardiac cells, and are caused by signaling defects. Consequently, modeling the electrical activity of the heart, and the complex signaling models that subtend dangerous arrhythmias such as tachycardia and fibrillation, necessitates a quantitative model of action potential (AP) propagation. Yet, many electrophysiological models, which accurately reproduce dynamical characteristic of the action potential in cells, have been introduced. However, these models are very complex and are very time consuming computationally. Consequently, a large amount of research is consecrated to design models with less computational complexity.

Results

This paper is presenting a new model for analyzing the propagation of ionic concentrations and electrical potential in space and time. In this model, the transport of ions is governed by Nernst-Planck flux equation (NP), and the electrical interaction of the species is described by a new cable equation. These set of equations form a system of coupled partial nonlinear differential equations that is solved numerically. In the first we describe the mathematical model. To realize the numerical simulation of our model, we proceed by a finite element discretization and then we choose an appropriate resolution algorithm.

Conclusions

We give numerical simulations obtained for different input scenarios in the case of suicide substrate reaction which were compared to those obtained in literature. These input scenarios have been chosen so as to provide an intuitive understanding of dynamics of the model. By accessing time and space domains, it is shown that interpreting the electrical potential of cell membrane at steady state is incorrect. This model is general and applies to ions of any charge in space and time domains. The results obtained show a complete agreement with literature findings and also with the physical interpretation of the phenomenon. Furthermore, various numerical experiments are presented to confirm the accuracy, efficiency and stability of the proposed method. In particular, we show that the scheme is second-order accurate in space.

     

Neuro-fuzzy model of homocysteine metabolism

In view of well-documented association of hyperhomocysteinaemia with a wide spectrum of diseases and higher incidence of vitamin deficiencies in Indians, we proposed a mathematical model to forecast the role of demographic and genetic variables in influencing homocysteine metabolism and investigated the influence of life style modulations in controlling homocysteine levels. Total plasma homocysteine levels were measured in fasting samples using reverse phase HPLC. Multiple linear regression (MLR) and neuro-fuzzy models were developed. The MLR model explained 64% variability in homocysteine, while the neuro-fuzzy model showed higher accuracy in predicting homocysteine with a mean absolute error of 0.00002 \(\mu \hbox {mol}/\hbox {L}\). Methylene tetrahydrofolate reductase (MTHFR) C677T, 5-methyltetrahydrofolate homocysteine methyltransferase (MTR) A2756G and 5-methyltetrahydrofolate homocysteine methyltransferase reductase (MTRR) A66G were shown to be positively associatiated with homocysteine, while nonvegetarian diet, serine hydroxymethyltransferase 1 (SHMT1) C1420T and TYMS \(5^\prime \)-UTR 28 bp tandem repeat exhibited negative association with homocysteine. The protective role of SHMT1 C1420T was attributed to more H-bonding interactions in the mutant modelled compared to the wild type, as shown through in silico analysis. To conclude, polymorphisms in genes regulating remethylation of homocysteine strongly influence homocysteine levels. The restoration of one-carbon homeostasis by SHMT1 C1420T or increased flux of folate towards remethylation due to TYMS \(5^\prime \)-UTR 28 bp tandem repeat or nonvegetarian diet can lower homocysteine levels.     

Flux theory for Poisson distributed pores with Gaussian permeability

The mean of the solute flux through membrane pores depends on the random distribution and permeability of the pores. Mathematical models including such randomness factors make it possible to obtain statistical parameters for pore characterization. Here, assuming that pores follow a Poisson distribution in the lipid phase and that their permeabilities follow a Gaussian distribution, a mathematical model for solute dynamics is obtained by applying a general result from a previous work regarding any number of different kinds of randomly distributed pores.

The new proposed theory is studied using experimental parameters obtained elsewhere, and a method for finding the mean single pore flux rate from liposome flux assays is suggested. This method is useful for pores without requiring studies by patch-clamp in single cells or single-channel recordings. However, it does not apply in the case of ion-selective channels, in which a more complex flux law combining the concentration and electrical gradient is required.     

On dependency properties of the ISIs generated by a two-compartmental neuronal model

One-dimensional leaky integrate and fire neuronal models describe interspike intervals (ISIs) of a neuron as a renewal process and disregarding the neuron geometry. Many multi-compartment models account for the geometrical features of the neuron but are too complex for their mathematical tractability. Leaky integrate and fire two-compartment models seem a good compromise between mathematical tractability and an improved realism. They indeed allow to relax the renewal hypothesis, typical of one-dimensional models, without introducing too strong mathematical difficulties. Here, we pursue the analysis of the two-compartment model studied by Lansky and Rodriguez (Phys D 132:267–286, 1999), aiming of introducing some specific mathematical results used together with simulation techniques. With the aid of these methods, we investigate dependency properties of ISIs for different values of the model parameters. We show that an increase of the input increases the strength of the dependence between successive ISIs.     

Transitory memory retrieval in a biologically plausible neural network model

A number of memory models have been proposed. These all have the basic structure that excitatory neurons are reciprocally connected by recurrent connections together with the connections with inhibitory neurons, which yields associative memory (i.e., pattern completion) and successive retrieval of memory. In most of the models, a simple mathematical model for a neuron in the form of a discrete map is adopted. It has not, however, been clarified whether behaviors like associative memory and successive retrieval of memory appear when a biologically plausible neuron model is used. In this paper, we propose a network model for associative memory and successive retrieval of memory based on Pinsky-Rinzel neurons. The state of pattern completion in associative memory can be observed with an appropriate balance of excitatory and inhibitory connection strengths. Increasing of the connection strength of inhibitory interneurons changes the state of memory retrieval from associative memory to successive retrieval of memory. We investigate this transition.     

Spike tranfer in statistical neuron assemblies. II. Neuron--nonlinear spike source]

The mathematical model of the neuron function is known to rely on space summing of excitement. The spikes contribute to the inner state of the neuron the farther from cell soma the synapses are located. The difference between excitatory and inhibitory effect results in spike firing if only neural firing threshold is achieved. The values of spike flux have been estimated on the basis of the model of CA3 sector of the Hippocampus and were found to be 15 divided by 35 imp/s.     

Mathematical model for cooperation of ion pump, symport, and antiport in epithelial cells

Transcellular transport in epithelial cells plays an important role in providing such physiological functions as excretion of cytotoxic substances or reabsorption of metabolites useful for the body life activity. These functions have been shown to be performed by the mechanisms - symport, antiport, ion pumps, and channels - that often function cooperatively. Kinetic models of the substrate transport with the aid of the above mechanisms are widely described in the literature. Much less attention is paid to modeling of cooperative activity of transporters that have different transport mechanisms. In this work we propose a mathematical model for flux coupling of three transporters - the ion pump, symporter, and antiporter as well as of two substrates, one of which (A) can be transported simultaneously by the symport and antiport mechanisms, while the other (B) - only by the latter mechanisms. Analysis of the model has shown that for the pair of substrates (A and B) the flux coupling becomes possible if the following conditions are met: (1) the substrate A flux into the internal cell volume using the symport mechanism is to exceed its antiporter-realized flux in the opposite direction; (2) probability of reorientation from one side of membrane to the other side for the antiporter loaded with the substrate is to be essentially higher than that for empty transporter. The proposed model can be used for comparing efficiency both of excretion and of reabsorption of cell metabolites in representatives of different taxa.     

Bone refilling in cortical basic multicellular units: insights into tetracycline double labelling from a computational model

Bone remodelling is carried out by ‘bone multicellular units’ ( $\text{ BMU }$ s) in which active osteoclasts and active osteoblasts are spatially and temporally coupled. The refilling of new bone by osteoblasts towards the back of the $\text{ BMU }$ occurs at a rate that depends both on the number of osteoblasts and on their secretory activity. In cortical bone, a linear phenomenological relationship between matrix apposition rate and $\text{ BMU }$ cavity radius is found experimentally. How this relationship emerges from the combination of complex, nonlinear regulations of osteoblast number and secretory activity is unknown. Here, we extend our previous mathematical model of cell development within a single cortical $\text{ BMU }$ to investigate how osteoblast number and osteoblast secretory activity vary along the $\text{ BMU }$ ’s closing cone. The mathematical model is based on biochemical coupling between osteoclasts and osteoblasts of various maturity and includes the differentiation of osteoblasts into osteocytes and bone lining cells, as well as the influence of $\text{ BMU }$ cavity shrinkage on osteoblast development and activity. Matrix apposition rates predicted by the model are compared with data from tetracycline double labelling experiments. We find that the linear phenomenological relationship observed in these experiments between matrix apposition rate and $\text{ BMU }$ cavity radius holds for most of the refilling phase simulated by our model, but not near the start and end of refilling. This suggests that at a particular bone site undergoing remodelling, bone formation starts and ends rapidly, supporting the hypothesis that osteoblasts behave synchronously. Our model also suggests that part of the observed cross-sectional variability in tetracycline data may be due to different bone sites being refilled by $\text{ BMU }$ s at different stages of their lifetime. The different stages of a $\text{ BMU }$ ’s lifetime (such as initiation stage, progression stage, and termination stage) depend on whether the cell populations within the $\text{ BMU }$ are still developing or have reached a quasi-steady state whilst travelling through bone. We find that due to their longer lifespan, active osteoblasts reach a quasi-steady distribution more slowly than active osteoclasts. We suggest that this fact may locally enlarge the Haversian canal diameter (due to a local lack of osteoblasts compared to osteoclasts) near the $\text{ BMU }$ ’s point of origin.     

Response of Isolated Rat Liver Mitochondria to Variation of External Osmolarity in KCl Medium: Regulation of Matrix Volume and Oxidative Phosphorylation

When isolated rat liver mitochondria are incubated in KCl medium, matrix volume, flux, and forces in both hypo- and hyperosmolarity are time-dependent. In hypoosmotic KCl medium, matrix volume is regulated via the K⁺/H⁺ exchanger. In hyperosmotic medium, the volume is regulated in such a manner that at steady state, which is reached within 4 min, it is maintained whatever the hyperosmolarity. This regulation is Pi- and -dependent, indicating Pi-K salt entry into the matrix. Under steady state, hyperosmolarity has no effect on isolated rat liver mitochondria energetic parameters such as respiratory rate, proton electrochemical potential difference, and oxidative phosphorylation yield. Hypoosmolarity decreases the NADH/NAD⁺ ratio, state 3 respiratory rate, and , while oxidative phosphorylation yield is not significantly modified. This indicates kinetic control upstream the respiratory chain. This study points out the key role of potassium on the regulation of matrix volume, flux, and forces. Indeed, while matrix volume is regulated in NaCl hyperosmotic medium, flux and force restoration in hyperosmotic medium occurs only in the presence of external potassium.     

Laser interferometry of the hydrolytic changes in protein solutions: the refractive index and hydration shells

Using an original laser interferometer of enhanced sensitivity, an increase in the refractive index of a protein solution was observed during the reaction of proteolysis catalyzed by pepsin. The increase in the refractive index of the protein solution at a concentration of 4 mg/ml was \( 9 \times 10^{-6} \) for bovine serum albumin and \(2.4 \times 10^{- 6}\) for lysozyme. The observed effect disproves the existing idea that the refractive index of protein solutions is determined only by their amino acid composition and concentration. It is shown that the refractive index also depends on the state of protein fragmentation. A mathematical model of proteolysis and a real-time method for estimating the state of protein hydration based on the measurement of refractive index during the reaction are proposed. A good agreement between the experimental and calculated time dependences of the refractive index shows that the growth of the surface of protein fragments and the change in the number of hydration cavities during proteolysis can be responsible for the observed effect.     

Calcium gating of H+ fluxes in chloroplasts affects acid-base-driven ATP formation

In previous work, calcium ions, bound at the lumenal side of the CF₀H⁺ channel, were suggested to keep a H⁺ flux gating site closed, favoring sequestered domain H⁺ ions flowing directly into the CF₀-CF₁ and driving ATP formation by a localized gradient. Treatments expected to displace Ca⁺⁺ from binding sites had the effect of allowing H⁺ ions in the sequestered domains to equilibrate with the lumen, and energy coupling showed delocalized characteristics. The existence of such a gating function implies that a closed-gate configuration would block lumenal H⁺ ions from entering the CF₀-CF₁ complex. In this work that prediction was tested using as an assay the dark, acid-base jump ATP formation phenomenon driven by H⁺ ions derived from succinic acid loaded into the lumen.Chlorpromazine, a photoaffinity probe for many proteins having high-affinity Ca⁺⁺-binding sites, covalently binds to the 8-kDa CF₀ subunit in the largest amounts when there is sufficient Ca⁺⁺ to favor the localized energy coupling mode, i.e., the gate closed configuration. Photoaffinity-bound chlorpromazine blocked 50% or more of the succinate-dependent acid-base jump ATP formation, provided that the ionic conditions during the UV photoaffinity treatment were those which favor a localized energy coupling pattern and a higher level of chlorpromazine labeling of the 8-kDa CF₀ subunit. Thylakoids held under conditions favoring a delocalized energy coupling mode and less chlorpromazine labeling of the CF₀ subunit did not show any inhibition of acid-base jump ATP formation.Chlorpromazine and calmidazolium, another Ca⁺⁺-binding site probe, were also shown to block redox-derived H⁺ initially released into sequestered domains from entering the lumen, at low levels of domain H⁺ accumulation, but not at higher H⁺ uptake levels; ie., the closed gate state can be overcome by sufficiently acidic conditions. That is consistent with the observation that the inhibition of lumenal succinate-dependent ATP formation by photoaffinity-attached chlorpromazine can be reversed by lowering the pH of the acid stage from 5.5 to 4.5.The evidence is consistent with the concept that Ca⁺⁺ bound at the lumenal side of the CF₀ H⁺ channel can block H⁺ flux from either direction, consistent with the existence of a molecular structure in the CF₀ complex having the properties of a gate for H⁺ flux across the inner boundary of the CF₀. Such a gate could control the expression of localized or delocalized energy coupling gradients.     

Coupling effects on turning points of infectious diseases epidemics in scale-free networks

Background

Pandemic is a typical spreading phenomenon that can be observed in the human society and is dependent on the structure of the social network. The Susceptible-Infective-Recovered (SIR) model describes spreading phenomena using two spreading factors; contagiousness (β) and recovery rate (γ). Some network models are trying to reflect the social network, but the real structure is difficult to uncover.

Methods

We have developed a spreading phenomenon simulator that can input the epidemic parameters and network parameters and performed the experiment of disease propagation. The simulation result was analyzed to construct a new marker VRTP distribution. We also induced the VRTP formula for three of the network mathematical models.

Results

We suggest new marker VRTP (value of recovered on turning point) to describe the coupling between the SIR spreading and the Scale-free (SF) network and observe the aspects of the coupling effects with the various of spreading and network parameters. We also derive the analytic formulation of VRTP in the fully mixed model, the configuration model, and the degree-based model respectively in the mathematical function form for the insights on the relationship between experimental simulation and theoretical consideration.

Conclusions

We discover the coupling effect between SIR spreading and SF network through devising novel marker VRTP which reflects the shifting effect and relates to entropy.

     

Coupling of Na<Superscript>+</Superscript>/α-ketoglutarate symport and PAH/α-ketoglutarate antiport in epithelial cells. Estimation of probability of the anion exchanger spontaneous re-orientation

In this work, dynamics was studied of uptake of p-aminohippurate by basolateral membrane vesicles isolated from rat kidney proximal tubules. The uphill PAH transport into the basolateral membrane vesicles was shown to occur in the presence of α-ketoglutarate and Na⁺-gradient. Based on mathematical model of symport and antiport cooperation, the mechanism of energy coupling of PAH transport via exchanger with Na⁺-dicarboxylate symport is discussed. Based on comparison of our own and literature data, the data analysis shows adequacy of the proposed mathematical model to describe the symport and antiport cooperation. This model has been shown to enable estimation of re-orientation probability of the empty anion exchanger (without substrate) from one membrane side to the other.     

Exploring how extracellular electric field modulates neuron activity through dynamical analysis of a two-compartment neuron model

To investigate how extracellular electric field modulates neuron activity, a reduced two-compartment neuron model in the presence of electric field is introduced in this study. Depending on neuronal geometric and internal coupling parameters, the behaviors of the model have been studied extensively. The neuron model can exist in quiescent state or repetitive spiking state in response to electric field stimulus. Negative electric field mainly acts as inhibitory stimulus to the neuron, positive weak electric field could modulate spiking frequency and spike timing when the neuron is already active, and positive electric fields with sufficient intensity could directly trigger neuronal spiking in the absence of other stimulations. By bifurcation analysis, it is observed that there is saddle-node on invariant circle bifurcation, supercritical Hopf bifurcation and subcritical Hopf bifurcation appearing in the obtained two parameter bifurcation diagrams. The bifurcation structures and electric field thresholds for triggering neuron firing are determined by neuronal geometric and coupling parameters. The model predicts that the neurons with a nonsymmetric morphology between soma and dendrite, are more sensitive to electric field stimulus than those with the spherical structure. These findings suggest that neuronal geometric features play a crucial role in electric field effects on the polarization of neuronal compartments. Moreover, by determining the electric field threshold of our biophysical model, we could accurately distinguish between suprathreshold and subthreshold electric fields. Our study highlights the effects of extracellular electric field on neuronal activity from the biophysical modeling point of view. These insights into the dynamical mechanism of electric field may contribute to the investigation and development of electromagnetic therapies, and the model in our study could be further extended to a neuronal network in which the effects of electric fields on network activity may be investigated.     

Temporal correlation based learning in neuron models

We study a learning rule based upon the temporal correlation (weighted by a learning kernel) between incoming spikes and the internal state of the postsynaptic neuron, building upon previous studies of spike timing dependent synaptic plasticity (Kempter, R., Gerstner, W., van Hemmen, J.L., Wagner, H., 1998. Extracting Oscillations: Neuronal coincidence detection with noisy periodic spike input. Neural computation 10, 1987–2017; Kempter, R., Gerstner, W., van Hemmen, J.L., 1999. Hebbian learning and spiking neurons. Physical Reviewm E59, 4498–4514; van Hemmen, J.L., 2001. Theory of synaptic plasticity. In: Moss, F., Gielen, S. (Eds.), Handbook of biological physics. vol. 4, Neuro Informatics, neural modelling, Elsevier, Amsterdam, pp. 771–823. Our learning rule for the synaptic weight w _ij is where the t _j,μ are the arrival times of spikes from the presynaptic neuron j and the function u(t) describes the state of the postsynaptic neuron i. Thus, the spike-triggered average contained in the inner integral is weighted by a kernel Γ(s), the learning window, positive for negative, negative for positive values of the time difference s between post- and presynaptic activity. An antisymmetry assumption for the learning window enables us to derive analytical expressions for a general class of neuron models and to study the changes in input-output relationships following from synaptic weight changes. This is a genuinely non-linear effect (Song, S., Miller, K., Abbott, L., 2000. Competitive Hebbian learning through spike timing dependent synaptic plasticity. Nature Neuroscience 3, 919–926).     

Electrical activity and metabolism in cardiac tissue: An experimental and theoretical study

Summary (1) Effects of the metabolic inhibitor 2,4-dinitrophenol (DNP) on electrical activity in frog atria were studied by means of the sucrose-gap technique and in tracer experiments. (2) Voltage-clamp studies of ionic membrane currents showed a suppression by DNP of peak Na inward current without marked changes in the kinetics of the Na-carrying system and an increase of steady state outward current to three to five times its normal value. In⁴²K tracer experiments, DNP increased K resting efflux by about 10% and decreased K influx by 25 to 30%. (3) The depression of Na inward current is regarded as being caused by a partial block of Na channels and an increase of internal Na concentration after inhibition of active Na extrusion. (4) The strong rise in outward current is probably not caused by a K current since K efflux fails to show a correspondingly large change. As a possible explanation for current and flux changes, an electrogenic K pump is discussed. (5) A mathematical model of a carrier system transporting a single ion species is described. The system is designed as a direct potential pump. Uphill transport requires an asymmetry of the rate constants governing the cyclic formation and breakdown of carrier-ion complex. The asymmetry is brought about by an input of metabolic energy. Reduction of energy input decreases the asymmetry and induces a carrier-mediated downhill ion movement, with corresponding changes in membrane current and ion fluxes. (6) A model of electrogenic K inward transport is calculated that approximately accounts for the steady state current and the K flux changes experimentally observed after inhibition.