Fractional calculus in bioengineering.

Richard L Magin

doi:10.1615/critrevbiomedeng.v32.i1.10

Home/Archive/PubMed

PubMed · 15248549

Fractional calculus in bioengineering.

Richard L Magin

View original source ↗Export metadata ↓

Abstract

Fractional calculus (integral and differential operations of noninteger order) is not often used to model biological systems. Although the basic mathematical ideas were developed long ago by the mathematicians Leibniz (1695), Liouville (1834), Riemann (1892), and others and brought to the attention of the engineering world by Oliver Heaviside in the 1890s, it was not until 1974 that the first book on the topic was published by Oldham and Spanier. Recent monographs and symposia proceedings have highlighted the application of fractional calculus in physics, continuum mechanics, signal processing, and electromagnetics, but with few examples of applications in bioengineering. This is surprising because the methods of fractional calculus, when defined as a Laplace or Fourier convolution product, are suitable for solving many problems in biomedical research. For example, early studies by Cole (1933) and Hodgkin (1946) of the electrical properties of nerve cell membranes and the propagation of electrical signals are well characterized by differential equations of fractional order. The solution involves a generalization of the exponential function to the Mittag-Leffler function, which provides a better fit to the observed cell membrane data. A parallel application of fractional derivatives to viscoelastic materials establishes, in a natural way, hereditary integrals and the power law (Nutting/Scott Blair) stress-strain relationship for modeling biomaterials. In this review, I will introduce the idea of fractional operations by following the original approach of Heaviside, demonstrate the basic operations of fractional calculus on well-behaved functions (step, ramp, pulse, sinusoid) of engineering interest, and give specific examples from electrochemistry, physics, bioengineering, and biophysics. The fractional derivative accurately describes natural phenomena that occur in such common engineering problems as heat transfer, electrode/electrolyte behavior, and sub-threshold nerve propagation. By expanding the range of mathematical operations to include fractional calculus, we can develop new and potentially useful functional relationships for modeling complex biological systems in a direct and rigorous manner.

Explore related subjects

Action Potentials Algorithms Animals Biomedical Engineering Cell Membrane Computer Simulation Connective Tissue Elasticity Humans Membrane Potentials Models, Biological Models, Neurological Neurons Viscosity Journal Article Review

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Richard L Magin. 2004. Fractional calculus in bioengineering.. https://doi.org/10.1615/critrevbiomedeng.v32.i1.10

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related citations

PubMed2007

Relationship of complex spike synchrony bands and climbing fiber projection determined by reference to aldolase C compartments in crus IIa of the rat cerebellar cortex.

Synchronous complex spike (CS) activity occurs most often among cerebellar Purkinje cells located in a narrow longitudinal (parasagittal) strip of cortex (synchrony band). The relationship of the anatomical organization of the olivocerebellar projection to these synchrony bands has not been investigated in detail. Thus, we studied this relationship by using the aldolase C (zebrin II) expression pattern, another landmark for the cerebellar longitudinal organization, as a reference frame in rat crus IIa. Crus IIa consists of 10 aldolase C-positive and -negative longitudinal compartments. Aldolase C labeling after multiple-electrode recording of CSs indicated that in lateral crus IIa (compartments 5+ to 7+) synchrony bands were generally constrained to single compartments. In contrast, in medial crus IIa (compartments 4a- to 5a-) the synchrony within and across the compartments was much higher than in lateral crus IIa, resulting in wide synchrony bands covering multiple compartments. Retrograde labeling of olivary neurons by injections of biotinylated dextran amine into aldolase C compartments in crus IIa showed that compartments in medial crus IIa were all innervated by the caudal part of the medial accessory olive. On the other hand, each aldolase C compartment in the lateral crus IIa was innervated by a region in a different subnucleus in the rostral inferior olive. These regions in different subnuclei were located close to each other. These results suggest that CS synchrony bands reflect the olivocerebellar compartmental projection pattern and neuronal coupling within a particular olivary subnucleus, and that medial and lateral crus IIa may be functionally distinct.

Action Potentials↗

PubMed2007

Motoneurons of the flight power muscles of the blowfly Calliphora erythrocephala: structures and mutual dye coupling.

The morphologies of the motoneurons of the dorsolongitudinal and the three dorsoventral flight power muscles (DLM, DVM 1-3) of Calliphora were investigated by means of cobalt backfills and intracellular biocytin stainings. The DLM is innervated by four prothoracic motoneurons supplying the four ventral muscle fibers and one mesothoracic motoneuron supplying the two dorsal fibers. The three fibers of the DVM 1 and the two fibers of the DVM 2 are innervated by five mesothoracic motoneurons, whereas the two fibers of the DVM 3 are innervated by two prothoracic motoneurons. In general, the motoneurons of each muscle have a common ventral soma cluster located in a characteristic position on the ipsilateral side of the thoracic ganglion, show similar dendritic arborizations in the mesothoracic wing neuropil, and have the same axon pathway. Only the soma of the common motoneuron of two dorsal fibers of the DLM is situated dorsally in the contralateral hemiganglion. The motoneurons of each muscle were found to be strongly dye coupled with each other, indicating that they are connected by gap junctions. In addition, the motoneurons of each muscle establish characteristic coupling patterns with the motoneurons of the other flight power muscles on both sides of the thorax and with two bilateral groups of local mesothoracic interneurons. The revealed coupling patterns are assumed to be of major relevance for the generation the characteristic, rhythmic flight activity of the motoneurons described in previous studies.

Action Potentials↗

PubMed2007

Systemic signalling in barley through action potentials.

Using apoplastic voltage- and ion selective microprobes, in barley leaves action potentials (APs) have been measured, which propagate acropetally as well as basipetally from leaf to leaf or from root to leaf following the application of mild salt stress (e.g. 30-50 mM KCl or NH(4)Cl) or amino acids (e.g. 1 mM glutamic acid or 5 mM GABA). Voltage changes were biphasic, followed an 'all-or-none' characteristic, and propagated at 20-30 cm min(-1) irrespective of the direction. With the salt-induced APs, a strong initial depolarization is the main AP-releasing factor that first causes Ca(2+) influx and then anion efflux. Ca(2+) influx coincides with an initial slower depolarization, the rapid anion efflux causes the typical voltage 'break-through'. Subsequently, K(+)-efflux starts after the depolarizing voltage has passed the K(+) equilibrium potential (inversion of the K(+) driving force). Glutamic acid and GABA induce APs not through membrane depolarization, but presumably by binding to a putative receptor or to ligand-gated Ca(2+)-conducting channels, respectively, followed by Ca(2+) induced activation of anion efflux. APs are accompanied by transient apoplastic pH increase (about 1 unit), and by cytoplasmic pH decrease (about 0.5 units). The apoplastic pH change is interpreted as an indicator of stress, the cytoplasmic pH change as a prerequisite for defence related gene activation. Since APs are released by agents added in a moderate concentration range, it is suggested that they may serve as first and fast systemic signals following attack from pathogens.

Action Potentials↗