Marion Silies

Christian Klämbt
email: klaembt@uni-muenster.de

APC/C Fzr/Cdh1 dependent regulation of cell adhesion controls glial migration in the Drosophila PNS

Keywords: Drosophila, glial cells, Fzr/Cdh1, APC/C, cell migration, Fasciclin 2 10; 2 movies

APC/CFzr/Cdh1 dependent regulation of cell adhesion controls glial migration in the Drosophila PNS

Marion Silies, Christian C Klambt

Introduction

Any complex nervous system is characterized by the precise placement of both, neurons and glial cells. Neuron-glia interaction is required in virtually all aspects of nervous system development and their accurate spatial patterning is key to proper brain function [START_REF] Nave | Axon-glial signaling and the glial support of axon function[END_REF][START_REF] Freeman | Glial cell biology in Drosophila and vertebrates[END_REF][START_REF] Halassa | Integrated brain circuits: astrocytic networks modulate neuronal activity and behavior[END_REF] . To establish their correct arrangement, many of the cells of the nervous system come to their final position by directed migration. This is especially true within the peripheral nervous system (PNS), where both, axonal growth cones and glial cells often navigate considerable distances before reaching their final destinations. In vertebrates and invertebrates, peripheral glial cells follow axonal projections into the periphery. They never overtake neuronal growth cones and strictly require preexisting axonal trajectories [START_REF] Freeman | Glial cell biology in Drosophila and vertebrates[END_REF][START_REF] Klambt | Modes and regulation of glial migration in vertebrates and invertebrates[END_REF][START_REF] Lyons | erbb3 and erbb2 are essential for Schwann cell migration and myelination in zebrafish[END_REF][START_REF] Gilmour | Migration and function of a glial subtype in the vertebrate peripheral nervous system[END_REF][START_REF] Kucenas | CNS-derived glia ensheath peripheral nerves and mediate motor root development[END_REF][START_REF] Sepp | Developmental dynamics of peripheral glia in Drosophila melanogaster[END_REF][START_REF] Silies | Glial cell migration in the eye disc[END_REF] . Thus, neuronal and glial migration must be tightly coordinated.

The PNS of the Drosophila larva develops during a short time window at mid embryogenesis. About 30 motor neurons are born in each abdominal segment and project their axons into the periphery. At about the same time a small set of sensory neurons sends their axons along the motor axons into the central nervous system (CNS). Only twelve uniquely identifiable glial cells distribute in a stereotypic manner along each peripheral nerve [START_REF] Sepp | Developmental dynamics of peripheral glia in Drosophila melanogaster[END_REF] . Seven of these glial cells are born in the CNS, the remaining are formed in the periphery [START_REF] Von Hilchen | Identity, origin, and migration of peripheral glial cells in the Drosophila embryo[END_REF] . Just after the first motor projections are established, the CNS born peripheral glial cells accumulate at the CNS-PNS transition zone and start their migration some time after ceasing to divide, suggesting that some inhibition first has to be released to allow migration.

Genetic screens have unravelled several factors required for migration and differentiation of Drosophila peripheral glial cells. As such, Notch and numb mutants show irregular glial cell migration with the leading edge cells displaying increased growth cones [START_REF] Edenfeld | Notch and Numb are required for normal migration of peripheral glia in Drosophila[END_REF] . Common cytoskeletal regulators, such as the small GTPases Rho and Rac, function in peripheral glial cell migration [START_REF] Sepp | RhoA and Rac1 GTPases mediate the dynamic rearrangement of actin in peripheral glia[END_REF] . Furthermore, mutants in the splicing factors Crooked neck and HOW interfere with glial differentiation and halt migration [START_REF] Edenfeld | The Splicing Factor Crooked Neck Associates with the RNA-Binding Protein HOW to Control Glial Cell Maturation in Drosophila[END_REF] . Whereas these genes are cell autonomously required in glial cells, nonautonomously acting functions regulating glial migration have been rarely described.

In gain of function experiments it was shown that the Netrin receptor Unc5 is able to direct the outwards migration of peripheral glial cells [START_REF] Freeman | Unwrapping glial biology: Gcm target genes regulating glial development, diversification, and function[END_REF] , but surprising few cells show migration defects in netrin or unc5 mutants [START_REF] Von Hilchen | Netrins guide migration of distinct glial cells in the Drosophila embryo[END_REF] . How the tight coupling between axonal and glial migration is coordinated, is not understood.

Here we describe a non-autonomous regulator of glial cell migration: In a genetic screen for components controlling migration of the peripheral glial cells we isolated several mutations in the cell cycle regulator Fizzy-related/Cdh1 (Fzr/Cdh1). We can genetically separate mitotic and postmitotic functions of Fzr/Cdh1 in glial development. Postmitotic Fzr/Cdh1 functions non-autonomously in neurons to control glial migration. We further identify the cell adhesion molecule Fasciclin2 as a critical downstream component. Adhesion between neurons and glia is mediated via a previously uncharacterized glial Fas2 isoform. Fzr/Cdh1 locally downregulates Fasciclin2 in the motor neuron, which is permissive for the initiation of glial cell migration. Our data lead to a model explaining how Fzr/Cdh1 modulates adhesion between neurons and glial cells and thus coordinates the onset of axonal development and glial cell migration.

Results

Identification of fzr/cdh1

To elucidate the mechanisms controlling glial migration, we characterized EMS mutants with defects in the peripheral glial pattern. In mutant 8F3, 19C6 and 20E1 embryos, the peripheral glia fail to initiate their migration and accumulate at the CNS/PNS transition zone (Fig. 1a,b; Suppl. Fig. 1a,b). Live imaging of migrating peripheral glial cells demonstrates a dramatic reduction in the extent of migration (Suppl. Movies 1,2). In addition, mutant embryos are characterized by an excess of glial cells, disrupted axonal tracts in the CNS and occasional axonal mistargeting in the PNS (Fig. 1a-c, Suppl. Fig. 1f,g). By deficiency mapping and complementation studies these mutants were identified as fizzy-related/cdh1 (fzr/cdh1) alleles.

Moreover, the fzr ie28 null allele [START_REF] Sigrist | Drosophila fizzy-related down-regulates mitotic cyclins and is required for cell proliferation arrest and entry into endocycles[END_REF] displays the same phenotypes (Fig. 1c), which can be rescued by ubiquitous expression of a fzr/cdh1 cDNA (not shown), confirming that fzr/cdh1 is essential for glial migration.

Based on the observed phenotypes, we identified additional fzr/cdh1 alleles in a collection of axon pattern mutants, formerly described as reduced axons (redax) alleles [START_REF] Hummel | Commissure formation in the embryonic CNS of Drosophila[END_REF] . Subsequent sequence analyses revealed mutations spanning the entire fzr/cdh1 coding sequence (M442, 8F3, G129, H089, 20E1) (Fig. 1e). Early stop codons in 20E1, G129 and H089 lead to truncated proteins, that can be detected by Western Blot analyses, whereas the shorter protein generated by the 8F3 mutation appears to be unstable (Suppl. Fig. 1c,d). The mutant 19C6 carries a Histidine to Leucine exchange in the third WD repeat (H255L) and shows elevated Fzr protein levels (Suppl. Fig. 1c,d).

Glial migration and proliferation defects are not linked

Fzr/Cdh1 is a well-known cell cycle regulator, which mediates arrest in G1 phase. In fzr/cdh1 mutants, mitotic cyclins accumulate and some epidermal cells undergo one additional round of mitosis [START_REF] Sigrist | Drosophila fizzy-related down-regulates mitotic cyclins and is required for cell proliferation arrest and entry into endocycles[END_REF] . Accordingly, the overall number of glial cells is increased from 35 glial cells in wild type hemineuromeres to 46 glial cells (Fig. 1a-c, Fig. 2h). Thus, fzr/cdh1 controls cell cycle arrest also in glial cells and this might explain the migratory deficits in fzr/cdh1 mutants.

To dissect a possible contribution of cell cycle defects for the migration phenotype we generated fzr/cdh1 cyclinA double mutants. cyclinA encodes an essential cell cycle regulator, which is upregulated in fzr/cdh1 mutants, but also inhibits Fzr/Cdh1 function [START_REF] Sigrist | Drosophila fizzy-related down-regulates mitotic cyclins and is required for cell proliferation arrest and entry into endocycles[END_REF][START_REF] Reber | Terminal mitoses require negative regulation of Fzr/Cdh1 by Cyclin A, preventing premature degradation of mitotic cyclins and String/Cdc25[END_REF] . cyclinA mutants show reduced glial cell numbers. In fzr/cdh1 cyclinA double mutant embryos the glial cell number is restored to wild type levels (Fig. 2ad,h). Interestingly, despite the rescue of their cell number, glial cells still fail to migrate in fzr/cdh1 cyclinA double mutants, indicating that fzr/cdh1 independently controls glial proliferation and migration (Fig. 2d,i). Furthermore, glial cells adopt a differentiated appearance in fzr/cdh1 mutants, as judged by the expression of GFP (Suppl. Fig. 2e,f).

Neuronal fzr/cdh1 controls glial cell migration

To further test whether the fzr/cdh1 controls glial migration postmitotically independent of its role in cell cycle, we conducted cell type specific rescue experiments. Expression of a wild type fzr/cdh1 cDNA in glial cells with repoGal4 or the even earlier acting gcmGal4 driver neither rescued glial cell number nor the migration defects (Fig. 2e,h,i and not shown), suggesting that fzr/cdh1 function is not required in glial cells. To assay the possible contribution of neuronal cells we expressed Fzr/Cdh1 under the control of the elavGal4 C155 enhancer trap. Whereas glial proliferation was not significantly affected (Fig. 2h), the migration phenotype was completely rescued in 49,6% of the hemisegments and partially rescued in 25,9% of the hemisegments (Fig. 2f,i; n=139; for definition of rescue strength see Suppl. Fig. 2a). This confirms that glial proliferation and migration can be independently addressed and shows that fzr/chd1 has a postmitotic, non-autonomous function in controlling glial migration.

When we employed an elavGal4 promoter fusion driver we noted an almost complete rescue of both the glial migration (71.5% full rescue; 19% partial rescue; n=105, Fig. 2g,i) and the glial proliferation phenotypes, confirming that this line confers some activity already in neuroblasts [START_REF] Berger | The commonly used marker ELAV is transiently expressed in neuroblasts and glial cells in the Drosophila embryonic CNS[END_REF] . Thus, fzr/chd1 can also rescue proliferation phenotypes, if expressed early enough. In addition, the axonal growth phenotypes in the CNS and the PNS are restored following fzr/cdh1 expression in the elav pattern (Fig. 2f,g, Suppl. Fig. 2b-d). In conclusion, our data show that fzr/cdh1 can rescue glial proliferation and migration independently. Interestingly, Fzr/Cdh1 functions postmitotically in neurons to control the movement of glial cells non-autonomously.

Glial migration requires APC/C function

Fzr/Cdh1 is a co-activator of the E3 ubiquitin ligase Anaphase Promoting Complex/Cyclosome (APC/C) (Suppl. Fig. 1e) [START_REF] Peters | The anaphase promoting complex/cyclosome: a machine designed to destroy[END_REF] . To test for APC/C activity in postmitotic neurons we expressed the well-described target protein CycB together with β-Galactosidase as expression control in motor neurons. Only little CycB accumulates in wild type whereas β-Gal is readily detectable, suggesting an efficient control of CycB protein levels via a functional APC/C (Fig. 3a,b). Consistent with this observation, overexpressed CycB accumulates in fzr/cdh1 mutant motor neurons, indicating that Fzr/Cdh1 can act through the APC/C in these cells (Fig. 3c,d).

In vitro experiments showed that binding of Fzr/Cdh1 to the APC/C requires the last two amino acids, the so-called IR tail [START_REF] Sigrist | Drosophila fizzy-related down-regulates mitotic cyclins and is required for cell proliferation arrest and entry into endocycles[END_REF][START_REF] Hummel | Commissure formation in the embryonic CNS of Drosophila[END_REF] . To determine whether Fzr/Cdh1 function is indeed mediated via the APC/C in the context of glial migration, we generated a fzr/cdh1 transgene lacking this APC/C binding motif (Fzr/Cdh1 ∆IR ) and compared its rescuing abilities to a wild type fzr/cdh1 construct. We employed PhiC31 mediated transgenesis to control for expression strength [START_REF] Reber | Terminal mitoses require negative regulation of Fzr/Cdh1 by Cyclin A, preventing premature degradation of mitotic cyclins and String/Cdc25[END_REF] . Fzr/Cdh1 ∆IR is able to rescue glial proliferation and migration but to a lower extend than wild type (Suppl. Fig. 3), indicating that APC/C binding is involved, but in vivo does not solely depend on the last two amino acids.

Mutants of the Drosophila homologues of several APC/C subunits (shattered/APC1, morula/APC2, ida/APC5, cdc27 and lemming/APC11) [START_REF] Peters | The anaphase promoting complex/cyclosome: a machine designed to destroy[END_REF] complete embryogenesis normally due to the persistence of maternal proteins and mRNA (not shown and Fig. 3e,f). Since the absolute requirement of the APC/C for cell division prevents the generation of germline clones, we generated zygotic double mutants of genes encoding the core subunits morula/APC2 and lemming/APC11. Such double mutants are embryonic lethal and display fzr/cdh1 like mutant glial phenotypes, which can be rescued by neural morula expression (Fig. 3g,h for quantification see Suppl. Fig. 4).

In summary, these data suggest that Fzr/Cdh1 non-autonomously controls glial migration via APC/C function in neurons.

Expression of Fzr/Cdh1 and APC/C in postmitotic neurons

Given the requirement of Fzr/Cdh1 in regulating glial migration, we next set out to analyze its expression. Efforts of several labs including ours generated antibodies that work in western blots, but fail to detect Fzr/Cdh1 protein in tissue. This might be due to very low levels of expression, as has also been noticed for the vertebrate homologue Cdh1 [START_REF] Gieffers | Expression of the CDH1-associated form of the anaphase-promoting complex in postmitotic neurons[END_REF] . In situ hybridization shows weak expression throughout the nervous system (Suppl. Fig. 5a-c). The P[lacZ] enhancer trap insertion in the fzr/cdh1 promoter region shows the same ubiquitous expression pattern and marks many postmitotic neurons (Fig. 4a). In line with the rescue data we noted only weak expression in glial cells (Fig. 4a).

Similarly, enhancer trap insertions in the lemming/APC11 and cdc27 revealed expression in many postmitotic neurons, including identified motor neurons (Suppl.

Fig. 5d-g).

To further investigate the subcellular localization of Fzr/Cdh1 we used transgenes directing the expression of GFP-or Myc-fusion proteins. These overexpressed proteins are functional as they can rescue fzr/cdh1 mutant phenotypes (Suppl. Fig. 5h and data not shown). Upon neuronal expression, the fusion proteins localize predominantly to the cytoplasm and the axons (Fig. 4b,c).

Furthermore, expression of a Morula/APC2-EGFP fusion labels axons and the cell soma (Suppl. Fig. 5i) raising the possibility that APC/C Fzr/Cdh1 could locally regulate axonal substrates inhibiting glial migration.

Fasciclin2 is a target of Fzr/Cdh1 function

Above we have shown that Fzr/Cdh1 acts non-autonomously in neurons to control glial migration via the APC/C. Thus, we reasoned that in fzr/cdh1 mutants, inhibitory axonal proteins accumulate to prevent glial migration. Consequently, the loss of this inhibitor in a fzr/cdh1 mutant background should suppress the glial migration phenotype. Following this logic, we designed a suppressor screen to identify new neuronal APC/C Fzr/Cdh1 substrates and additionally tested candidate molecules previously linked to ubiquitination and/or motor neuron differentiation as potential downstream targets. Double mutant analyses revealed the cell adhesion molecule Fasciclin2 (Fas2) as a specific interactor of Fzr/Cdh1 in peripheral glial migration. fas2 single mutants do not display any gross defects in the final positioning of the peripheral glial cells (Fig. 4e).

Interestingly, however, in embryos lacking both fzr/cdh1 as well as fas2, glial cell migration is substantially rescued (Fig. 4d-g, normal migration in 54,7%, partial migration in 27,9%, n=86). In such double mutants, the glial cell number is still increased and the axonal growth defects are as in fzr/cdh1 mutants (Fig. 4f,h,i), however, the peripheral glial cells distribute nicely along the peripheral nerves. This shows that glial migration phenotypes are not a secondary consequence of axonal phenotypes and that Fzr/Cdh1 acts through the adhesion molecule Fas2 to control glial migration. Mutations in the closely related adhesion protein Neuroglian (the Drosophila L1 homolog), which is expressed on glial and neuronal cells, do not suppress the glial migration phenotype seen in fzr/cdh1 mutants (Suppl. Fig. 6), indicating that the interaction between Fzr/Cdh1 and Fas2 is specific. In summary, out data suggest that Fzr/Cdh1 controls glial cell migration by regulating Fasciclin2 and that too much Fas2 mediated adhesion can prevent migration.

Neurons and glial cells express different Fas2 isoforms

Fzr/Cdh1 controls glial cell migration in the PNS via Fasciclin2, which is a commonly used motor neuron marker in Drosophila [START_REF] Grenningloh | Genetic analysis of growth cone guidance in Drosophila: fasciclin II functions as a neuronal recognition molecule[END_REF] . Thus the question arises, how the presence of the Fas2 protein on neurons is communicated to the glial cells. Fas2 is evolutionary conserved and its mammalian homolog NCAM or its Manduca homolog exist in different isoforms, of which the transmembrane bound variants are expressed by neurons, whereas a GPI linked isoform is found on glial cells [START_REF] Maness | Neural recognition molecules of the immunoglobulin superfamily: signaling transducers of axon guidance and neuronal migration[END_REF][START_REF] Wright | Cell type-specific expression of fasciclin II isoforms reveals neuronal-glial interactions during peripheral nerve growth[END_REF][START_REF] Higgins | Different isoforms of fasciclin II are expressed by a subset of developing olfactory receptor neurons and by olfactory-nerve glial cells during formation of glomeruli in the moth Manduca sexta[END_REF] .

The Drosophila fas2 gene had been described to encode three isoforms: two transmembrane isoforms (PEST+ and PEST-) and a GPI anchored version of the protein (Flybase, [START_REF] Grenningloh | Genetic analysis of growth cone guidance in Drosophila: fasciclin II functions as a neuronal recognition molecule[END_REF] ). We identified a fourth variant by RT-PCR (Fig. 5a,b, data not shown). The monoclonal antibody generally used to detect the Fas2 protein (mAb 1D4), recognizes two of these isoforms (Fas2 PA+ and Fas2 PA-) and serves as a motor neuron marker (Fig. 5a-c). However, an enhancer trap insertion in the fas2 promoter region suggests that in Drosophila, fas2 is expressed in a much wider pattern.

Besides expression in clusters of motor neurons, the enhancer trap also marks peripheral glial cells (Fig. 5d). To detect the localization of the different Fasciclin 2 proteins we generated several exon trap insertion lines which allow the visualization of distinct Fas2 isoforms even in living animals (Fig. 5a, e-g). Fas2 GFP397 tags the isoforms PA(PEST+), PA(PEST-) and PB (Fig. 5a). As observed in the enhancer trap, Fas2 GFP397 shows a broad expression. As expected for the PA isoforms, it highlights motor axons in the PNS, but additionally marks peripheral glial cells (Fig. 5e-g). This expression can therefore be assigned to the Fas2 PB isoform. Stationary glial cells at the CNS/PNS boundary exhibit prominent GFP-expression whereas low levels of Fas2 are found on migrating glial cells (Fig. 5e-h). This finding suggests that Fas2 can act as a homophilic adhesion protein between axons and glial cells.

Low Fasciclin2 levels allow initiation of glial migration

Since Fzr/Cdh1 functions non-autonomously in neurons to control migration, we revisited motor neuronal Fas2 expression. Careful re-examination of the distribution of the axonal Fas2 PA+/-isoforms during neurogenesis revealed a graded expression within the axon (Fig. 6a-d). The formation of this Fas2 gradient coincides with the onset of glial cell migration in stage 14. The graded distribution of Fas2 was quantified by measuring the ratio of expression levels behind the growth cone and at the CNS/PNS transition zone (ratio: 2.04, n=55 nerves) (Fig. 6i, Suppl. Fig. 7). This graded Fas2 expression is also observed in live embryos using the exon trap fas2 , which only labels the PA+ Fas2 protein variant (ratio 1.77, n=40 nerves, Suppl. Fig. 7). Interestingly, in most segments of fzr/cdh1 mutants this graded Fas2 expression is lost (ratio: 1.17, n=51 nerves, p < 0.01) (Fig. 6e-i). In agreement with this we noted a slight increase of the overall levels of Fas2 expression in mutant embryos (Suppl. Fig. 7j). Together with the phenotypic rescue in the fas2 fzr/cdh1 double mutant, this observation argues that the fzr/cdh1 glial phenotype is caused by local removal and thus, a change in the axonal distribution of Fas2.

Fas2 and its vertebrate homolog NCAM act as homophilic adhesion proteins and might modulate migration via adhesion or signalling activities [START_REF] Grenningloh | Genetic analysis of growth cone guidance in Drosophila: fasciclin II functions as a neuronal recognition molecule[END_REF][START_REF] Prag | NCAM regulates cell motility[END_REF][START_REF] Cavallaro | N-CAM modulates tumour-cell adhesion to matrix by inducing FGF-receptor signalling[END_REF][START_REF] Paratcha | The neural cell adhesion molecule NCAM is an alternative signaling receptor for GDNF family ligands[END_REF] . Our data predict that accumulation of Fas2 should stall peripheral glial migration. Indeed, overexpression of intact Fas2 or a transmembrane Fas2 lacking the cytoplasmic domain in glial cells and neurons in wild type embryos attenuates glial movement, whereas overexpression of a transmembrane bound Fas2 lacking the extracellular domain does not have any phenotypic effects (Fig. 6j-l, for quantification see Suppl. Fig. 8).

These latter findings also suggest that migration is regulated through differential adhesion and does not require Fas2 mediated signalling (Suppl. Fig. 9).

The graded expression of Fasciclin2 requires endocytosis

In conclusion, we demonstrate that the coordination of glial migration along its axonal substrates is regulated by APC/C Fzr/Cdh1 dependent stability of the Fas2 adhesion protein (Suppl. Fig. 9). Vertebrate NCAM can be targeted for endocytosis by ubiquitination in vitro [START_REF] Diestel | NCAM is ubiquitylated, endocytosed and recycled in neurons[END_REF] . Likewise, Fas2 recycling has been shown at the Drosophila neuromuscular junction [START_REF] Mathew | Drosophila amphiphysin functions during synaptic Fasciclin II membrane cycling[END_REF] . In agreement with these findings, we find a dotted cellular distribution of Fas2 protein within the CNS (Fig. 7a, Suppl. Fig. 10a-c), which in part colocalizes with different endocytic markers (Rab4, Rab5 or Rab11, Fig. 7b-d).

Furthermore, in fzr/cdh1 mutant embryos the amount of vesicular Fas2 expression is reduced (Suppl. Fig. 10d-f) and, moreover, when we blocked endocytosis by expressing a temperature-sensitive dominant negative Dynamin (UASshi ts ) or interfered with the endocytic pathway by expressing a dominant negative Rab4 construct in neurons, we phenocopied the glial migration phenotype of fzr/cdh1 mutants while the axonal scaffold stayed largely intact and concomitantly affect the graded Fas2 expression (Fig. 7e-j). Although we do not have evidence for a direct interaction of Fzr/Cdh1 and Fas2, our data are in line with a model that removal of Fas2 from the plasma membrane via endocytosis creates a graded distribution of axon-glia adhesion, which enables glial migration.

Discussion

Within the nervous system, glial cells generally show extensive migratory abilities [START_REF] Klambt | Modes and regulation of glial migration in vertebrates and invertebrates[END_REF] . This is in particular evident in the peripheral nervous system of Drosophila, where glial cells closely follow axons to eventually differentiate. Here we present a mechanism by which neurons determine the onset of glial migration and thus coordinate the development of these two cell types. We describe a new non-autonomous requirement of Fzr/Cdh1, one of the two main regulators of APC/C function [START_REF] Peters | The anaphase promoting complex/cyclosome: a machine designed to destroy[END_REF] , in non-autonomously controlling glial migration by regulating cell adhesion between neurons and glial cells (Suppl. Fig. 9). APC/C-Fzr/Cdh1 has been initially identified based on its central role in cell cycle control, but postmitotic APC/C function is also required during later stages of neuronal development [START_REF] Juo | The anaphase-promoting complex regulates the abundance of GLR-1 glutamate receptors in the ventral nerve cord of C. elegans[END_REF][START_REF] Van Roessel | Independent regulation of synaptic size and activity by the anaphase-promoting complex[END_REF] . We could genetically separate mitotic and postmitotic requirements of Fzr/Cdh1 function in glial development. Glial proliferation can be rescued without simultaneously rescuing migration and vice versa, glial migration can be specifically rescued independent of proliferation or axonal growth phenotypes.

In mice, APC/C Fzr/Cdh1 regulates axonal growth by regulating nuclear substrates [START_REF] Konishi | Cdh1-APC controls axonal growth and patterning in the mammalian brain[END_REF][START_REF] Stegmuller | Cell-intrinsic regulation of axonal morphogenesis by the Cdh1-APC target SnoN[END_REF] .

Interestingly, fzr/cdh1 regulates axonal growth in the opposite manner in Drosophila.

In addition, the Drosophila APC/C also appears to control synaptic size and activity 33 . Besides regulating different substrates, the other APC/C coactivator Cdc20 also participates in presynaptic development [START_REF] Yang | A Cdc20-APC ubiquitin signaling pathway regulates presynaptic differentiation[END_REF] . Thus, reiterative use of variations of the same ubiquitin ligase machinery allows control of many aspects of nervous system development.

In the context of glial cell migration, we identified Fasciclin2 as a critical downstream target of APC/C Fzr/Cdh1 . Within the Drosophila PNS, the glia navigates along the motor axons and requires a graded distribution of the homophilic adhesion protein Fasciclin2 to determine the onset of migration. This graded distribution is brought about by endocytosis. Although we were unable to demonstrate a direct ubiqutination of Fas2, NCAM is targeted for endocytosis via ubiquitination [START_REF] Diestel | NCAM is ubiquitylated, endocytosed and recycled in neurons[END_REF] . Cell adhesion molecules have frequently been linked to migration and migratory processes often require the dynamic modulation of adhesive properties. As such, the differential localization of adhesion proteins to different axonal domains has been described [START_REF] Dodd | Spatial regulation of axonal glycoprotein expression on subsets of embryonic spinal neurons[END_REF][START_REF] Bastiani | Expression of fasciclin I and II glycoproteins on subsets of axon pathways during neuronal development in the grasshopper[END_REF] . A subcellular gradient of the L1 family member Neurofascin is required for the correct GABAergic innervation of the axon initial segment of Purkinje neurons and high levels of Neurofascin are needed to stop the extension of the basket axon and to initiate synapse formation [START_REF] Ango | Ankyrin-based subcellular gradient of neurofascin, an immunoglobulin family protein, directs GABAergic innervation at purkinje axon initial segment[END_REF] . Another example stems from primary cell cultures of Drosophila neurons, in which the axonal membrane can be subdivided into distinct domains. Cell intrinsic properties, which in part require Dynamin dependent membrane recycling, establish the polarized distribution of membrane proteins [START_REF] Katsuki | Intra-axonal patterning: intrinsic compartmentalization of the axonal membrane in Drosophila neurons[END_REF] . Likewise, we showed that interference with the endocytic pathways disrupts both, the subcellular Fas2 distribution and glial cell migration. Furthermore, our data present an upstream mechanism how such a subcellular distribution of adhesiveness can be brought about and it will be interesting to determine whether the same or similar mechanism act in other contexts.

Although the local downregulation of Fas2 is critical to initiate glial migration, and specifically rescues fzr/cdh1 mutant glial migration phenotypes, fas2 mutants show a normal final positioning of glial cells. Interestingly, the modulation of Fas2 adhesiveness appears to be a more general concept underlying the control of migration. In the vertebrate CNS, adhesiveness mediated by the Fas2 homologue NCAM is altered by the addition of polysialic acid (PSA) [START_REF] Rutishauser | Polysialic acid in the plasticity of the developing and adult vertebrate nervous system[END_REF][START_REF] Weinhold | Genetic ablation of polysialic acid causes severe neurodevelopmental defects rescued by deletion of the neural cell adhesion molecule[END_REF] . Whereas NCAM mutants are viable, fertile and have only minor nervous system phenotypes [START_REF] Cremer | Inactivation of the N-CAM gene in mice results in size reduction of the olfactory bulb and deficits in spatial learning[END_REF] , loss of the two polysialyltransferases that transfer PSA to NCAM severely affects brain development, which in turn can be rescued by loss of NCAM [START_REF] Rutishauser | Polysialic acid in the plasticity of the developing and adult vertebrate nervous system[END_REF][START_REF] Weinhold | Genetic ablation of polysialic acid causes severe neurodevelopmental defects rescued by deletion of the neural cell adhesion molecule[END_REF] . In the context of Schwann cell migration, the vertebrate counterpart of the peripheral wrapping glial cells, GDNF can stimulate motility by decreasing NCAM mediated adhesiveness [START_REF] Paratcha | The neural cell adhesion molecule NCAM is an alternative signaling receptor for GDNF family ligands[END_REF] .

Whereas the concept of regulating adhesion to control migration seems conserved, the underlying mechanisms can be different.

While downregulation of axonal Fas2 is necessary to initiate migration, additional mechanisms must be in place to later attenuate glial movement, which in the Drosophila PNS could for example include mechanical contact to the peripheral born glial cells, which migrate ventrally [START_REF] Von Hilchen | Identity, origin, and migration of peripheral glial cells in the Drosophila embryo[END_REF][START_REF] Edenfeld | Notch and Numb are required for normal migration of peripheral glia in Drosophila[END_REF] . Such a contact dependent spacing of glial cells has been observed for oligodendrocyte progenitors in zebrafish development [START_REF] Kirby | In vivo time-lapse imaging shows dynamic oligodendrocyte progenitor behavior during zebrafish development[END_REF] .

Several isoforms of Fasciclin2 or its vertebrate homolog NCAM are generated in vivo, which show a conserved cell type specific expression pattern. GPI-linked Fas2 or NCAM proteins are expressed by glia, whereas the large transmembrane forms are found on neuronal membranes [START_REF] Wright | Different isoforms of fasciclin II play distinct roles in the guidance of neuronal migration during insect embryogenesis[END_REF] . Interestingly, these adhesion proteins not only control the degree of adherence between cell types, but can also function as regulators of cell signaling. As such, Fasciclin2 can inhibit EGF receptor signaling in the fly [START_REF] Mao | Fasciclin 2, the Drosophila orthologue of neural celladhesion molecule, inhibits EGF receptor signalling[END_REF] . Moreover it was demonstrated that NCAM, whose expression can be further regulated by the Ca 2+ dependent adhesion protein E-cadherin, affects fibroblast growth factor receptor (FGFR) function during cell migration [START_REF] Cavallaro | N-CAM modulates tumour-cell adhesion to matrix by inducing FGF-receptor signalling[END_REF][START_REF] Lehembre | NCAM-induced focal adhesion assembly: a functional switch upon loss of E-cadherin[END_REF] . Receptor tyrosine kinases have also been implicated in glial differentiation [START_REF] Franzdottir | Switch in FGF signalling initiates glial differentiation in the Drosophila eye[END_REF] . Therefore the analysis of Fas2 might serve as one entry point to decipher the link between glial migration along neuronal processes and the subsequent extension of glial membranes around the axon. e) Re-expression of fzr/cdh1 in all glial cells in a fzr/cdh1 ie28 mutant using the repoGal4 driver does not affect the glial phenotype. f) Re-expression of fzr/cdh1 in a fzr/cdh1 ie28 mutant directed by elavGal4 C155 rescues the glial cell migration phenotype but does not rescue the glial cell number. The quantification is shown in (h,i). g) Reexpression of fzr/cdh1 in a fzr/cdh1 ie28 mutant directed by a elavGal4 driver construct on the third chromosome results in an almost complete phenotypic rescue of both the glial and the axonal phenotypes. h, i) Quantification of the results obtained in the different cell type specific rescue experiments. Error bars show the standard deviation, n represents the number of analyzed embryos, the number of hemisegments is given in parentheses. Statistical analyses were performed with a Welch t-test. Significant differences of the mean are indicated by asterisks (p<0.01, 99% confidence). Scale bar is 20 μm. to 34°C look like wild type, glial migration is disrupted in elav>>shi ts 1 expressing embryos at 34°C. h,i) Stage 14 embryos marked with anti-Fas2 (false color representing intensity). In comparison to control embryos from the same staining (h), there is a weaker gradient of Fas2 expression in embryos expressing a dominant negative Rab4 in neurons (i). j) Quantification of the distal/proximal-ratio of Fas2 expression in control and elav>>rab4 SN embryos. Plot details as in Figure 6.

Statistical analyses was performed with a Welch t-test. (p<0.01, 99% confidence).

Scale bar is 5 μm in (b-d) and 20 μm (a, e-i).

Methods

Histology

Embryos were staged and stained according to standard protocols. Flies were allowed to lay eggs at 25° and embryos were aged at 18°C or 25°C. For expression of the temperature sensitive UASshi ts 1, embryos were aged at 34°C. Antibodies used: mouse anti-Repo 1:5, mouse anti-Fas2 1:5 (1D4), mouse anti-Myc 1:5 (9E10) and anti-Futsch 1:20 (22C10) (DSHB); rabbit anti-ßGalactosidase (Cappel) 1:1000; rabbit anti-GFP 1:1000 (Invitrogen) and anti-HRP-Cy5 or anti-HRP-Cy3 (Dianova).

Secondary antibodies were from Dianova. Specimens were analyzed on a Zeiss 510 or 710 LSM.

Statistical analysis

For quantification of the Fas2 expression levels, we defined two regions of interest close to the CNS / PNS transition zone (see supplementary Figure 7h for an example) and determined the mean intensity using the ImageJ software (NIH).

Statistical analysis was performed using R (http://www.r-project.org/).

Live cell imaging of glial cell migration

Wild type or fzr ie28 stage 13-14 embryos homozygous for repo4.3Gal4

UASstingerGFP were oriented on an agar plate and transferred to a cover slip coated with heptane parcel tape glue. Embryos were covered in Voltalef halocarbon oil (Sigma-Aldrich) and imaged on a Spinning disc (Ultra View RS, Perkin Elmer) in 1 μm sections with a rate of 1 z-stack/min. Data were processed with the Image J software (NIH).

Fly strains and genetics

fzr 8F3 , fzr 20E1 , fzr 19C6 were isolated in a screen for peripheral glia migration phenotypes [START_REF] Edenfeld | Notch and Numb are required for normal migration of peripheral glia in Drosophila[END_REF][START_REF] Edenfeld | The Splicing Factor Crooked Neck Associates with the RNA-Binding Protein HOW to Control Glial Cell Maturation in Drosophila[END_REF] , fzr redax alleles have been identified in a screen for CNS axon pattern defects [START_REF] Hummel | Commissure formation in the embryonic CNS of Drosophila[END_REF] . The duplication Dp(1;Y)dx+5 was used to rescue fzr mutants and fzr ie28 [START_REF] Sigrist | Drosophila fizzy-related down-regulates mitotic cyclins and is required for cell proliferation arrest and entry into endocycles[END_REF] , Df(1)JC70 and fzr G0418 were used for subsequent complementation tests (Bloomington stock centre and C. Lehner). For initial rescue experiments, repoGal4, gcmGal4, elav(III)Gal4, elav C155 Gal4, daGal4 and UASfzr (C. Lehner) were used. Subsequently, we generated φC31 based transgenes in the 86Fb landing site (UASfzr ΔIR , UASfzr wt ) [START_REF] Bischof | An optimized transgenesis system for Drosophila using germ-line-specific phiC31 integrases[END_REF] . The fas2 exon trap insertion fas2 GFP397 was isolated in an exon trap screen, the Fas2 GFPCB03613 exon trap insertion was obtained from the Spradling lab [START_REF] Buszczak | The carnegie protein trap library: a versatile tool for Drosophila developmental studies[END_REF] . Other stocks: fas2 EB112 20

DNA work and transgenes

φC31 based transgenes were made by generating a pENTR clone of the entire fzr coding sequence from the cDNA clone RE20929 and subsequent Gateway cloning into pUAST-attB-rfA. For the UASfzr ΔIR construct, the last two amino acids were deleted. Both constructs were injected into flies carrying the same landing site (86Fb). A N-terminally myc-tagged Fzr protein was generated by Gateway cloning from the above described pENTR-fzr into pUAST-myc-rfA (pTMW) and injected into w 1118 flies using standard procedures. A EGFP Morula fusion was cloned in pUAST attB rfA EGFP. Details are available upon request.

Western blot analyses and sequencing of fzr alleles

For Western blot analyses, 50 stage 16 embryos were collected in 35μl SDS-sample buffer, homogenized, boiled and 15μl were loaded on a 10% SDS gel. The blot was probed with rabbit antiFzr (1:3000, C. Lehner). Sequencing of all mutant alleles was performed on heterozygous DNA from adult flies.

Figure 1

 1 Figure 1 Identification of fzr/cdh1 mutants

Figure 2

 2 Figure 2 Postmitotic Fzr/Cdh1 non-autonomously regulates glial cell migration

Figure 3

 3 Figure 3 Postmitotic APC/C functions in motor neurons to control glial migration

Figure 4

 4 Figure 4 Fzr/Chd1 regulates glial cell migration via the homophilic cell adhesion molecule Fasciclin 2

Figure 5 Broad expression of fasciclin 2 a)

 52 Figure 5 Broad expression of fasciclin 2

Figure 6

 6 Figure 6 Fzr/Cdh1 regulates the graded expression of Fasciclin 2 at the onset of glial migration

Figure 7

 7 Figure 7 Disruption of the endocytic pathway disrupts glial migration and reduces the graded expression of Fasciclin 2 a) In a stage 16 CNS, Fas2 marks the longitudinal axon tracks. In addition, many vesicle-like structures can be detected in the regions where motor neurons reside. Regions of the CNS as indicated by the box are shown in (b-d). b-d) αFas2 (magenta) labels UASrab11 GFP , UASrab5 GFP or UASrab4 GFP expressing vesicles in neurons (green, arrowheads). The image on the right shows only the Fas2 expression. e-g) Disruption of the endocytic pathway in neurons leads to glial migration defects. Frontal views of dissected Drosophila stage 16 nervous system preparations. Neuronal membranes are shown in green (HRP), glial cell nuclei in magenta (Repo), the genotype is indicated. e, f) Whereas elavGal4 embryos shifted

  , UAS myc fzr, UASEGFP::Mr/APC2 (this work), UASactGFP and P[lacW]fas2 rG272 , P[lacW]cdc27 L7123 , UAScycB, UASshi ts (T. Clandinin), Mz97Gal4, fas2Gal4 Mz507 (B. Altenhein), UASfas2 intra YFP, UASfas2 extra YFP, UASfas2 intact YFP (A. Nose), UASrab4 DN (G. Tear), UASstingerGFP, Df(2R)Chig[230], lmg EY11317 , lmg 03424 , cdc27 L712 , fas2 rG272 (Bloomington stock center) and ftz ng20 Gal4 UASflp tub>CD2>Gal4 2x UAS myr mRFP (M. Landgraf).

Acknowledgements

We thank C. Lehner, J. Raff, B. Altenhein, T. Orr-Weaver, G. Tear, K. Basler, G. Tear, M. Landgraf and A. Nose for sending flies and antibodies. T. Hummel, E. Raz, A. Püschel and T.R. Clandinin for comments on the manuscript, G. Edenfeld, K. Krukkert, N. Otto, A. Weinbrecht and H. Neuert for help with initial experiments, J. Lipp for help with statistical analysis and members of the Klämbt lab for continuous support and discussions. This work was supported by a PhD fellowship of the Boehringer Ingelheim Foundation to M.S. and grants of the Deutsche Forschungsgemeinschaft to C.K..

In fzr/cdh1 mutants, all sensory neuronal clusters, but fewer axons are present. g')

shows the boxed area from g), imaged with the same settings than wild type (f). (h,i) CNS of stage 16 embryos marked with HRP (green) and Hb9 (magenta). (h',i') Single channels. Motor neuronal cell number does not appear increased in fzr/cdh1 mutants. the graded expression of Fas2 (green) vs. the motor neuronal membrane marker myr mRFP (red). g) Quantification of the Fasciclin2 gradient in living embryos expressing Fas2 GFP CB03613 , which labels only the Fas2 PA+ isoform (see Figure 5a).

Supplementary

Box-Whisker plot represents the median (bold line), 25% and 75% quartiles (box), 1,5 times interquartile distance (whiskers) and extreme values (circles). For quantification of the steepness of the gradient of Fas2 PA+ and Fas2 PA-in fixed wild type and fzr/cdh1 mutant embryos see Figure 6. h,i) The ratio of the mean intensities of Fas2 expression was defined as the ratio of the staining intensity in a distal region of the nerve (d, behind the growth cone) and a proximal region (p, CNS-PNS transition zone). h) Fas2 expression in green (h) or false color (i, ZEN software, ZEISS). j) For western blot analysis, stage 16 fzr/cdh1 8F3 , fas2 GFP CB03613 hemizygous mutant embryos (+/Y) and heterozygous controls (+/-) were selected. In comparison to the control, total Fas2 GFP CB03613 levels are slightly reduced in whole embryo lysates from fzr 8F3 mutants.

Supplementary

Supplementary Figure 9 Model

Model underlying glial migration. At the onset of glial cell migration, APC/C Fzr/Cdh1 is required to locally downregulate Fas2 (red) so that glial cells can initiate their journey into the periphery. In fzr/cdh1 mutants, the motor neurons present too high levels of Fas2, which prevents glial cell migration.

Supplementary

Supplementary Movie 1

Wild type embryo carrying two repoGal4 and two UAS-GFP stinger transgenes. Anterior is up, the CNS is to the left. The movie was generated on a spinning disc microscope. One z-scan was taken every minute over a 120 min interval.

Supplementary Movie 2

fzr/cdh1 ie28 mutant type embryo carrying two repoGal4 and two UAS-GFP stinger transgenes. Anterior is up, the CNS is to the left. The movie was generated on a spinning disc microscope. One z-scan was taken every minute over a 120 min interval.

Supplementary Information Titles

Supplementary Item

& Number

Title or Caption

Supplementary Figure 1 Characterization of the fzr/cdh1 phenotype Supplementary Figure 2 Classification of glial migration, axonal growth and glial differentiation Supplementary Figure 3 Structure function analysis of Fzr/Cdh1

Supplementary Figure 4 The APC/C is required for glial cell migration Supplementary Supplementary Video 1 Wild type embryo carrying two repoGal4 and two UAS-GFP stinger transgenes. Anterior is up, the CNS is to the left. The movie was generated on a spinning disc microscope. One z-scan was taken every minute over a 120 min interval. Supplementary Video 2 fzr/cdh1 ie28 mutant type embryo carrying two repoGal4 and two UAS-GFP stinger transgenes. Anterior is up, the CNS is to the left. The movie was generated on a spinning disc microscope. One z-scan was taken every minute over a 120 min interval.


Carla M A Pinto

J A Tenreiro Machado

Fractional Central Pattern Generators for Bipedal Locomotion

Locomotion has been a major research issue in the last few years. Many models for the locomotion rhythms of quadrupeds, hexapods, bipeds and other animals have been proposed. This study has also been extended to the control of rhythmic movements of adaptive legged robots.

In this paper, we consider a fractional version of a central pattern generator (CPG) model for locomotion in bipeds. A fractional derivative

, with α non-integer, is a generalization of the concept of an integer derivative, where

The integer CPG model has been proposed by Golubitsky, Stewart, Buono and Collins, and studied later by Pinto and Golubitsky. It is a network of four coupled identical oscillators and has dihedral symmetry. We study parameter regions where periodic solutions, identified with legs' rhythms in bipeds, occur, for

We find that the amplitude and the period of the periodic solutions, identified with biped rhythms, increase as α varies from near 0 to values close to one.

.

ℵ ∈ n

These networks consist of n 4 coupled identical cells, arranged in two cycles of n 2 cells, that are able to generate the phase relations seen in the animal gaits. These phase relations are a result of the symmetry of the model. The model presents the bilateral symmetry of animals and a cyclic translational symmetry, from back to front, that is, cell n 4 is coupled to cell 2 , and the same applies for cells 1 4n

Introduction

A CPG is a network of neurons located somewhere in the nervous system capable of producing the rhythms associated to different animal gaits [START_REF] Grillner | Locomotion in vertebrates: central mechanisms and reflex interaction[END_REF]. Mathematically, CPGs are modeled as networks of identical systems of differential equations, where the individual systems model (collections of) neurons (or cells) (Collins & Stewart, 1993a;[START_REF] Collins | Hexapodal gaits and coupled nonlinear oscillator models[END_REF][START_REF] Collins | A group-theoretic approach to rings of coupled biological oscillators[END_REF][START_REF] Kopell | Coupled oscillators and the design of central pattern generators[END_REF][START_REF] Golubitsky | A modular network for legged locomotion[END_REF][START_REF] Golubitsky | Symmetry in locomotor central pattern generators and animal gaits[END_REF]. These CPG models have been studied in animal locomotion and, more recently, in the field of robotics. [START_REF] Golubitsky | A modular network for legged locomotion[END_REF][START_REF] Golubitsky | Symmetry in locomotor central pattern generators and animal gaits[END_REF] propose models for locomotor CPGs for n 2 -legged animals, for all values of and 1 . Similar network architectures also generate locomotor patterns of legless animals, such as worms, snakes and lampreys. In those locomotor networks, the term "leg" should be interpreted as "muscular unit". Unwanted conjugacies presented in previous CPG models for animal gaits, such as the one between trot and pace in the four-cells CPG models for quadrupeds [START_REF] Collins | Coupled nonlinear oscillators and the symmetries of animal gaits[END_REF], are prevented by this new network architecture. These new structures for CPG models are derived assuming that each joint receives signals from two neurons. This reasoning draws an analogy with muscular action in real joints, since two muscle groups, flexors and extensors, control most joints. Thus, abstractly, CPG models control muscle groups rather than legs and minimal CPG networks should have two cells per leg.

In the field of robotics, most models for CPG locomotor patterns of robots have one oscillator per joint, and consequently, each joint is driven by two neurons [START_REF] Taga | Self-organized control of bipedal locomotion by neural oscillators in unpredictable environment[END_REF]Liu et al, 2008;Baydin, 2008). There are a few authors though that, similarly to [START_REF] Collins | Coupled nonlinear oscillators and the symmetries of animal gaits[END_REF], assume that locomotor CPGs have only one cell per limb (Righetti et al, 2006;[START_REF] Righetti | Pattern generators with sensory feedback for the control of quadruped locomotion[END_REF]. The main disadvantage, as explained previously for the four-cells CPG networks of Collins and Stewart, is that it is necessary to change the architecture of the network in order to observe all of the animal movements in the robot. The advantage of this approach is that only half of the oscillators are needed to control the motor patterns, comparatively with the CPG networks proposed by Golubitsky et al [START_REF] Golubitsky | A modular network for legged locomotion[END_REF][START_REF] Golubitsky | Symmetry in locomotor central pattern generators and animal gaits[END_REF], and, therefore, less parameter values need to be recomputed in gait transition.

CPG-like coupled cell systems are able to produce sustained rhythmic activation patterns even when isolated from external stimuli. CPGs have other nice properties, such as limit cycle behaviour, phase locking modes, robustness against small perturbations and smooth online modulation of trajectories. All these characteristics can be achieved through changes in parameter values of the equations [START_REF] Marder | Central pattern generators and the control of rhythmic movements[END_REF][START_REF] Matos | A Brainstem-like Modulation Approach for Gait Transition in a Quadruped Robot[END_REF], Righetti et al, 2006). These features make CPGs an attractive option to model the control of legged robot locomotion for bipedal, quadrupedal and other designs [START_REF] Marder | Central pattern generators and the control of rhythmic movements[END_REF][START_REF] Matos | A Brainstem-like Modulation Approach for Gait Transition in a Quadruped Robot[END_REF], Righetti et al, 2006[START_REF] Righetti | Pattern generators with sensory feedback for the control of quadruped locomotion[END_REF]. In Matos et al [START_REF] Matos | A Brainstem-like Modulation Approach for Gait Transition in a Quadruped Robot[END_REF], it is proposed a bio-inspired robotic controller able to generate locomotion and to easily switch between different types of gaits. The duty factor and inter limb phase values are adjusted according to the velocities. This improves stability and response of the robot during locomotion. Degallier et al [START_REF] Degallier | Movement generation using dynamical systems: a humanoid robot performing a drumming task[END_REF] use a dynamical systems' approach yielding the online generation trajectory in a robot performing a drumming task. These trajectories have both rhythmic and discrete parts. Righetti et al [START_REF] Righetti | Pattern generators with sensory feedback for the control of quadruped locomotion[END_REF] designed a CPG architecture based on coupled dynamical systems where gait transition is achieved by parameter change and where sensory feedback is included. Righetti and Ijspeert (Righett & Ijspeert, 2006) use symmetric coupled systems theory [START_REF] Golubitsky | Nonlinear dynamics of network: the groupoid formalism[END_REF] to design a generic coupling architecture for a CPG and showed how it could be applied to the control of a crawling humanoid robot.

To conclude, it is also demonstrated that CPG-like controllers can be successfully implemented as analog electronic circuits [START_REF] Lewis | An in silico central pattern generator: silicon oscillator, coupling, entrainment, and physical computation[END_REF][START_REF] Tenore | Entrainment of silicon central pattern generators for legged locomotory control[END_REF].

The network architecture LEG for the central pattern generator for bipedal legs rhythms studied here is shown in Fig. 1. The CPG network LEG consists of four coupled identical cells. In the graph of Fig. 1, the nodes represent the cells and the arrows represent the couplings. Different arrows mean different coupling strengths. This network is capable of producing periodic solution types identified with biped locomotion rhythms, namely, walk, run, two-legged jump, two-legged hop, hesitation-walk, asymmetric hop, skip, gallop, and one-legged hop. In this paper we consider a fractional version of the CPG LEG represented in Fig. 1. We vary the order of α and we analyze the amplitude and period changes of periodic solutions produced by the fourcell CPG model and identified with legs' rhythms in bipeds.

2.-Fractional calculus -summary

Fractional calculus (FC) is a generalization of the ordinary integer differentiation and integration to an arbitrary order [START_REF] Oldham | The fractional calculus: theory and application of differentiation and integration to arbitrary order[END_REF][START_REF] Samko | Fractional integrals and derivatives: theory and applications[END_REF][START_REF] Miller | An introduction to the fractional calculus and fractional differential equations[END_REF]. The subject was initiated in 1695 by Leibniz that sent a letter to L'Hospital with the question: "Can the meaning of derivatives with integer order be generalized to derivatives with non-integer orders?". In the last two decades we witnessed an increasing interest in the FC and relevant applications emerged in the areas of physics and engineering (Outstaloup, 1991;[START_REF] Mainardi | Fractional relaxation-oscillation and fractional diffusion-wave phenomena[END_REF][START_REF] Machado | Analysis and design of fractional-order digital control systems[END_REF][START_REF] Nigmatullin | The statistics of the fractional moments: Is there any chance to "read quantitatively" any randomness?[END_REF][START_REF] Podlubny | Fractional-order systems and PI D -controllers[END_REF][START_REF] Machado | Discrete-time fractional-order controllers[END_REF][START_REF] Chen | Discretization schemes for fractional-order differentiators and integrators[END_REF][START_REF] Baleanu | About fractional quantization and fractional variational principles[END_REF][START_REF] Machado | Fractional Derivatives: Probability Interpretation and Frequency Response of Rational Approximations[END_REF].

There are several definitions of fractional derivatives, being three of the most important the Riemann -Liouville, the Grunwald -Letnikov, and the Caputo given by:

( ) ( ) ( ) ( ) n n d t a n t f n dt n d n t f t D a < < - ∫ + - - - Γ = α τ α τ τ α α 1 , 1 1 (1) ( ) ( ) ( ) ∑ - = - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - → = ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ h a t k kh t f k k h h t f t D a 0 1 1 0 lim α α α (2) ( ) ( ) ( ) ( ) ( ) n n d t a n t n f n t f t D a < < - ∫ + - - - Γ = α τ α τ τ α α 1 , 1 1 (3)
where ( ) Γ is the Euler's gamma function, [ ] x means the integer part of x, and h is the step time increment.

It is also possible to generalize several results based on transforms, yielding expressions such as the Laplace expression:

( ) { } ( ) { } ( ) + - - ∑ - = - = 0 1 0 1 0 0 f k t D n k k s t f L s t f t D L α α α (4)
where s and L represent the Laplace variable and operator, respectively.

These definitions demonstrate that fractional derivatives capture the history of the variable, or, by other words, have memory, contrary to integer derivatives, that are local operators.

The Grunwald -Letnikov formulation inspires the numerical calculation of the fractional derivative based on the approximation of the time increment h through the sampling period T and the series truncation at the rth term. This method is often denoted as Power Series Expansion (PSE) yielding the equation in the z -domain:

( ) { } ( ) ( ) ( ) ( ) z X r k k z k k k T t x D Z ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∑ = - + - Γ + Γ - ≈ 0 1 ! 1 1 1 α α α α (5)
where ( )

( ) { } t x Z z X =
and z and Z represent the z-transform variable and operator, respectively.

In fact, expression (2) represents the Euler (or first backward difference) approximation in the z s → discretization scheme, being the Tustin approximation another possibility. The Euler and Tustin rational expressions, ( ) ( )

1 1 1 1 0 - - = - z T z H and ( ) 1 1 1 1 2 1 1 - + - - = - z z T z H
respectively, are often called generating approximants of zero and first order, respectively. Therefore, the generalization of these conversion methods leads to the non-integer order results:

( ) ( ) 1 0 1 1 1 - = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ - - ≈ z H z T s α α α (6.a) ( ) 1 1 1 1 1 1 2 - = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - + - - ≈ z H z z T s α α α (6.b)
We can obtain a family of fractional differentiators generated by ( ) , yielding:

1 0 - z H α and ( ) 1 1 - z H
( ) ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - - + ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - ≈ 1 1 1 1 0 z H p z pH s α α α (7)
In order to get a rational expression, the final approximation corresponds to a PSE or a rational fraction expansion. This approach is often denoted by Continued Fraction Expansion (CFE) of order

ℵ ∈ k
, based on a Padé expansion in the neighborhood of

0 1 = - z , yielding: ( ) ∑ = - ∑ = - = - k i i z i b k i i z i a z k H 0 0 1 , ℜ ∈ i b i a , (8)
Since one parameter is linearly dependent, usually it is established that

1 0 = b .

CPG model for bipeds -review

In this chapter we review the work of Pinto and Golubitsky [START_REF] Pinto | Central Pattern Generators for Bipedal Locomotion[END_REF] for the CPG model for legs rhythms in bipeds. We state the general class of differential equations corresponding to LEG and we review the so-called K H / theory [START_REF] Golubitsky | The Symmetry Perspective[END_REF] that helps to identify periodic solutions produced by CPG LEG with known biped locomotion patterns.

.CPG LEG -equations and symmetries

The class of ordinary differential equations (ODEs) governing the CPG model in Fig. 1 is the following:

) 1 , , , ( ) , , , ( ) , , , ( ) , , , ( 2
3 4 4 2 1 4 3 3 3 4 1 2 2 4 3 2 1 1 x x x x F x x x x x F x x x x x F x x x x x F x = = = = (9)
where

k i x ℜ ∈ is the cell 4 , 3 , 2 , 1 = i variables, ℵ ∈ k
is the dimension of the cells' internal dynamics, and

( ) k k F ℜ → ℜ 4 :
is an arbitrary mapping. As all cells are assumed to be identical then F must be the same for all cells. The coupled cell system in (9) has 2 D -symmetry. Consider the transposition that switches muscle groups between legs, we define it as

) 34 )( 12 ( = ρ
. The other permutation that the coupled system (9) allows is the switch between muscle groups in each leg. Define the later as

). 24 )( 13 ( = τ
These two independent permutations generate the symmetry group of order 4, denoted as 2 D and given by:

) ( 2 ) ( 2 2 τ ρ Z Z D × = (10)
This CPG model LEG is capable of producing eight periodic solutions corresponding to the bipedal gaits of two-legged hop, two-legged jump, walk, run, one-legged hop, hesitation walk, gallop, skip, and asymmetric hop. See Pinto and Golubitsky [START_REF] Pinto | Central Pattern Generators for Bipedal Locomotion[END_REF], for the reasoning of this identification.

. The

K H / theory

The identification of the periodic solutions of the network model LEG with biped leg rhythms is done using symmetry arguments. The K H / theorem [START_REF] Golubitsky | The Symmetry Perspective[END_REF] uses the symmetry group of a network to enumerate those pairs of spatiotemporal symmetry subgroups H and K that can correspond to periodic solutions of differential equations associated to the network. These solutions are then identified with known biped locomotion patterns.

Let Γ be a finite group of symmetries of the following systems of ODEs:

) (x f x = (11) Let ) (t x
be a periodic solution of system (11), with period normalized to 1, and let

Γ ∈ γ
. By symmetry, we know that

) (t x γ
is also a solution to (11). We consider two cases, either both solutions intersect (thus, are identical) or the two trajectories do not intersect. If both solutions are identical, then by uniqueness of solutions there is

1 S ∈ θ such that ), ( ) ( θ γ - = t x t x or ), ( ) ( t x t x = + θ γ
and we say that

) (t x has spatiotemporal symmetry Γ ∈ γ
, where θ is the temporal phase shift that corresponds to . γ If 0 = θ then γ is a spatial symmetry.

Define K as the subgroup of all spatial symmetries and H as the subgroup of all spatiotemporal symmetries. Mathematically, we have:

{ } { } { } { } t t x t x H t t x t x K ∀ = Γ ∈ = ∀ = Γ ∈ = , ) ( ) ( : ), ( ) ( : γ γ γ γ (12)
In what follows, we discuss how symmetries of periodic solutions ( H and K ), in the CPG model LEG, can correspond to the biped gaits of walk and run. A similar reasoning applies to the other bipedal gaits.

Walk and run are gaits in which the two legs move half period out-of-phase. Nevertheless, these are different gaits. In the walk, muscle groups in each leg move like a pendulum and in the run, these muscles move in unison [START_REF] Mann | Biomechanics. In Disorders of the Foot[END_REF][START_REF] Mann | Comparative electromyography of the lower extremity in jogging, running and sprinting[END_REF]). These differences can be explained by their symmetry group pairs ( ).

, K H

Let τ be the permutation that switches muscle groups between legs in bipeds. Applying τ to the run does not change that gait since muscles inside each leg receive the same signal. Applying τ to a walk does change the gait, in the sense that the muscle groups inside each leg receive signals that are now a half-period out of phase from the signals they received before. The later implies that the two legs are also half-period out of phase from where they were before applying .

τ The permutation τ is called a spatial symmetry for the run (the phase shift is zero) and is called a spatiotemporal symmetry for the walk (the phase shift is 2 1 ). More precisely, applying τ to a walk and then performing a half-period phase shift does not change the walk gait. Consider now the permutation .

ρτ Applying ρτ to a walk does not change the gait, since pairs of diagonal muscles receive the same signal and the two pairs of signals are still half-period out of phase. Whereas applying symmetry ρτ to a run results in the same gait but shifted by one half-period. Thus, ρτ is a spatial symmetry of the walk and a spatiotemporal symmetry of the run.

For H and K to correspond to symmetries of periodic solutions of CPG LEG, there are some algebraic properties to be satisfied, namely, K H / must be a cyclic group [START_REF] Pinto | Central Pattern Generators for Bipedal Locomotion[END_REF], and references therein).

To conclude, we note that the K H / theory may be used to infer the CPG structure, for locomotion of n 2 -legged animals, from the desired types of periodic solutions. Moreover, [START_REF] Buono | Models of central pattern generators for quadruped locomotion I. Primary gaits[END_REF] have used this mathematical tool to prove that there is only one CPG model with eight cells capable of producing periodic solution types identified with the quadruped walk, trot and pace.

Results and discussion

We simulate the fractional version of the coupled cell system (9), given by:

) 1 , , , ( 4 )
, , , ( 3 )
, , , ( 2 )
, , , ( 2 3 4 2 1 4 3 3 4 1 2 4 3 2 1 1 x x x x F x D x x x x F x D x x x x F x D x x x x F x D = = = = α α α α (13)
where α D , with α non-integer, is a generalization of the concept of an integer derivative, where

. 1 = α
We consider the Morris-Lecar equations [START_REF] Morris | Voltage Oscillations in the Barnacle Giant Muscle Fiber[END_REF] as internal cell dynamics. The coupling is linearly diffusive. The nondimensionalized Morris-Lecar equations [START_REF] Rinzel | Analysis of neural excitability and oscillations[END_REF] are a system of two ordinary differential equations given by:

( )( ) ( ) ( ) ( ) ( ) ( ) w v, g w - n(v) (v) w w v, f i + k v - v w k g - l v - v l g - 1 - v v m Ca g v = = = - = φτ (14) where . 4 2 3 cosh ) ( , 4 3 tanh 1 2 1 ) ( , 2 1 tanh 1 2 1 ) ( ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - + = ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ - + = v v v v v v v v n v v v v m τ Each term ( )( ) 1 - v v m Ca g , ( ) l v - v l g and ( ) k v - v w k g -
models an ionic channel that regulates the voltage v along the membrane of the axon. Parameter i is the current stimulus.

We adopt the PSE method for the approximation of the fractional derivative in the discrete time numerical integration. However, several experiments demonstrated that it was required a slight adaption to the standard approach based on a simple truncation of the series. In fact, since our objective is to generate limit cycles, the truncation corresponds to a diminishing of the gain (Tenreiro [START_REF] Machado | Fractional Derivatives: Probability Interpretation and Frequency Response of Rational Approximations[END_REF] and, consequently, leads to difficulties in the promotion of periodic orbits. Therefore, in order to overcome this limitation, we decided to include a gain adjustment factor corresponding to the sum of the truncated series coefficients. In this line of thought, since the sum of the coefficients from [START_REF] Machado | Fractional Derivatives: Probability Interpretation and Frequency Response of Rational Approximations[END_REF], the fractional derivative approximation adopted in the sequel is:

1 = k up to ∞ = k must be 1.0 (Tenreiro
( ) { } ( ) ( ) ( ) ( ) ( ) ( ) ( ) z X r k k z k k k T r k k k k t x D Z ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ∑ = - + - Γ + Γ - - ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ∑ = + - Γ + Γ - ≈ 0 1 ! 1 1 1 1 1 1 ! 1 1 α α α α α α 3 3 3 3 2 3 32 1 3 12 4 3 21 3 3 3 3 2 2 2 2 3 2 31 4 2 11 1 2 22 2 2 2 2 1 1 1 1 4 1 31 3 1 11 2 1 21 1 1 1 1 k y k x g k y k y k x k x k k x k x k k x k x k y k x f t k x H k x k y k x g t k y k y k x k x k k x k x k k x k x k y k x f t k x H k x k y k x g t k y k y k x k x k k x k x k k x k x k y k x f t k x H k x k y k x g t k y k y k x k x k k x k x k k x k x k y k x f k x H k x t t α α α α α α α α Δ Δ + = + - - - - - - Δ + = + Δ + = + - - - - - - Δ + = + Δ + = + - - - - - - Δ + = + Δ + = + - - - - - - + = + (16)
where

ℜ ∈ ij k
are the coupling constants and t Δ is the time increment.

The parameter values of the Morris-Lecar equations, in the numerical simulations, are chosen to be

, 2 . 0 , 3 . 0 , 4 . 0 , 1 . 0 4 3 2 1 = = = = v v v v , 8 . 1 , 6 . 0 = = k l g g 8 . 0 , 8 . 1 = - = k l v v and . 1 = i
In Tables 1 and2, we show initial conditions and parameter values for the periodic orbits, identified with the biped gaits of walk (Figures 234), two-legged jump (Figures 567), and run . Each simulation was executed until a stable periodic solution was found. Moreover, it was considered We compute the amplitude and the period of the solutions of system ( 16), for each value of α leading to Figures 111213141516. We find that, for all gaits, both amplitude and period values increase as α goes from near 0 to values close to 1. We also observe that, in the case of the periodic solution identified with the walk, the interval where the amplitude is greater than zero is larger than the interval for the other two gaits. This is a result of the initial condition used to simulate the corresponding periodic solution. We simulate another periodic solution identified with the walk, for an initial condition of the same order than those used here for the run and for the two-legged jump (Table 1). We find that the new interval, for amplitude values greater than zero, coincides with the one for the other two gaits. Another remark is that there is a slight increase in the values of the amplitude and the period as α goes to values close to 1. This is a numerical effect due to the series truncation in the approximation of the fractional derivative. As we can observe from Figures 11-16, α behaves like a bifurcation parameter, that is, as we vary α the dynamical behaviour of the system changes. This parameter, in particular, simplifies the work of the researcher. In fact, we do not need to know deeply the parameters of the equations, that model the dynamics of each cell, in order to increase the frequency or the amplitude of a given periodic solution, we just have to vary the value of . α

In Figure 17, we depict the 3D plot of the first variable of cell 1 dynamics and its derivative, for 80000 samples of the periodic solution of Figure 7.

Acknowledgments

This paper was supported (in part) by Fundação para a Ciência e a Tecnologia (FCT) through the Centro de Matemática da Universidade do Porto www.fc.up.pt/cmup.

Fig. 1 .

 1 Fig. 1. CPG network LEG for the control of biped legs. Cells 1 and 3 send signals to the left leg, cells 2 and 4 send signals to the right leg (Fig 2, Pinto & Golubitsky, 2006)

Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .Fig. 7 .Fig. 8 .Fig. 9 .Fig. 10 .Fig. 11 .

 234567891011 Fig. 2. Periodic solution of the CPG network LEG identified with the walk, for . 95 . 0 = α

Fig. 13 .

 13 Fig. 13. Amplitude of the periodic solution produced by the CPG network LEG identified with the two-legged jump, for different values of α .

Fig. 15 .

 15 Fig. 15. Amplitude of the periodic solution produced by the CPG network LEG identified with the run, for different values of α .

Fig. 17 .

 17 Fig. 17. 3D plot of 1x and its derivative, for 80000 samples, for the periodic solution of Figure7(two-legged jump).

Table 1 . Initial conditions for the bipedal gaits of walk, two-legged jump, and run, where

 1

	(	x	i	( ) ( ) ) k y k i ,

represents the initial condition of cell i at sample .

	walk								;x	=	.		x	=	.	;x	=	.
		y	=	.	;y	=	.		;y	=	.		;y	=	.	;y	=	.	;
		x	=	.	;x	=	.		;x	=	.		;x	=	.	;x	=	.	;
		y	=	.	;y	=	.	;y	=	.	;y	=	.	;y	=	.	;
		x	=	.	;x	=	.		;x	=	.		;x	=	.	;x	=	.	;
		y	=	.	;y	=	.	;y	=	.	;y	=	.	;y	=	.	;
		x	=	.	;x	=			;x	=	.	x	=	.	;x	=	.
		y	=	.	;y	=	.		;y	=	.	;y	=	.	;y	=	.	;
																	k
	Gait	Initial conditions

Table 2 . Parameter values for the bipedal gaits of walk, two-legged jump, and run, where t Δ is the time increment, and ij k are the coupling constants. For more information, see text.

 2

	Gait	t Δ	φ		g	Ca	11 k	12 k	k	21	k	22		k	31	k	32
	walk	0005 . 0	2 . 0	. 1	5	1 . 0	1 . 0	3 . 0 -	. 0 -	3	2 . 0	2 . 0
	two-	0005 . 0	. 1	0	. 1	8	1 . 0	1 . 0	5 . 0 -	. 0 -	5	. 0 -	8	8 . 0 -
	legged
	jump
	run	0005 . 0	. 1	0	. 1	8	1 . 0	1 . 0	0 . 0	0 . 0		. 0 -	7	7 . 0 -

 (15)The dynamics governing the fractional CPG model LEG is given by: