Diabetic neuropathy is one of the symptoms of prolonged hyperglycemia. It causes numbness and weakness in the legs, particularly in cats.
In diabetic neuropathy, nerves in the legs are progressively damaged, leading to tingliness, pain, numbness, and weakness or paralysis.
Symptoms are less common in dogs, but do occur. Many dogs have hind leg weakness when they are first diagnosed with diabetes. This is sometimes the symptom which brings the pet to the vet's office. It's often falsely attributed to "getting older" by the caregiver. What is also known as "diabetic neuropathy".  most often affects both rear legs and will progress symmetrically. With treatment and regulation, most dogs also have reversal of neuropathy 
This 1983 JAVMA abstract  indicates that both the neuropathy and low blood pressure returned to normal following the control of the diabetes with insulin.
Mechanics of diabetic neuropathyEditIn humans, the excess glucose oxidizes the tiny capillaries that nourish nerve cells, and the resulting constriction starves the nerve cells. (See Wikipedia:Diabetic Neuropathy link below. ) In dogs and cats the mechanism  seems different: the nerve axons dwindle and atrophy, and the spaces between them become clogged with glycogen deposits.
Understanding how this works, means some learning about nerve cells, impulses and their relationship to muscles. An Atlantic Coast Veterinary Conference (ACVC) 2001 presentation of feline neurological diseases  has a section dealing with diabetes-mediated neuropathy. The basic problems regarding nerves, muscles and neuropathy apply to all those with diabetes, so let's take it apart:
The most common neuropathy noted in the feline is diabetic neuropathy (secondary to diabetes mellitus). Clinical signs occur most commonly in the middle to older aged feline and present as lower neuron  signs (plantigrade stance)  in the hind limbs. Patellar (of the knee)  reflexes may be diminished, although flexor  reflexes and pain sensation are usually intact.
Muscles and nerves can't communicate properlyEdit
The etiology and pathogenesis of diabetic neuropathy involves primary and secondary axonal  (nerve fiber) degeneration due to slowing of axon  transport (Neuron impulses move along axons very much like electricity through a wire . Flaws in this communication system between nerves and muscles are the basis for neuropathy and other nerve-related disorders, such as Muscular Distrophy, etc.) secondary to hyperglycemia, alteration of macromolecular transport, interference with axon maintenance  and repair by the neuronal cell body, and neural hypoxia  (Insufficient oxygen--in this case to the neuron).
A gradual stateEdit
Progression to this state may occur over several months. Diagnosis and treatment is through the establishment of underlying disease (diabetes mellitus) and characteristic neurologic exam findings, and in most cases will resolve in 6 to 12 months if the DM is treated appropriately. Those who have a disease or diseases of the endocrine system (medical catch-all term is endocrinopathy) are more prone to neuropathic problems than anyone who has no endocrine conditions. 
Peripheral and other neuropathiesEdit
There is a form of neuropathy which has been known to doctors treating people with diabetes since the advent of insulin therapy. The name given to it is insulin neuritis, as it often occurs shortly after starting insulin.  The problem is that even though it's been around about as long as insulin treatment, the phenomenon is currently not considered common.  Insulin neuritis can follow the rapid improvement of diabetic control, but its mechanisms are still elusive  As you see from this link, one theory is that insulin neuritis is caused by the regeneration of the nerve axons. 
- ↑ 1.0 1.1 1.2 1.3 Diabetic neuropathy. Southpaws.com (Fall 1999).
- ↑ Diabetes Mellitus in Dogs and Cats. NAVC (January 2008).
- ↑ Johnson CA, Kittleson MD, Indrieri RJ. (1983). Peripheral Neuropathy & Hypotension in a Diabetic Dog. Journal-American Veterinary Medical Association.
- ↑ Church, David B. (2009). A Logical Approach to Weakness and Seizures. WSAVA.
- ↑ Wikipedia:Diabetic neuropathy
- ↑ Dahme E, Hafner A, Reusch C, Schmidt P. (1989). Diabetic Neuropathy in Dogs & Cats. Tierartzlichen Praxis (Veterinary Practice) in English.
- ↑ Kline, Karen (2001). Feline Neurologic Disease. Atlantic Coast Veterinary Conference.
- ↑ Biology Pages: Neuron.
- ↑ Plantigrade Stance. Drs. Foster & Smith Pet Education Library.
- ↑ 10.0 10.1 Biology Pages: Muscle Spindles.
- ↑ Biology Pages: Neurons.
- ↑ 12.0 12.1 Biology Pages: Neuromuscular Junction.
- ↑ Biology Pages: Excitable Cells.
- ↑ Biology Pages: Fueling Muscle Contraction.
- ↑ Biology Pages: Muscles.
- ↑ Peripheral Neuropathy. National Diabetes Information Clearinghouse (NDIC).
- ↑ Neuropathy Affects Nerves Throughout the Body. National Diabetes Information Clearinghouse (NDIC).
- ↑ Diabetic Neuropathies. Endotext.org.
- ↑ Under-recognised Paradox of Neuropathy from Rapid Glycemic Control. Postgraduate Medical Journal-Register--Free to Read (2004).
- ↑ Tesfaye, S., et. al. (1996). Arterio-venous Shunting and Proliferating New Vessels in Acute Painful Neuropathy of Rapid Glycemic Control (Insulin Neuritis). Diabetologica.
- ↑ Honma H, Podratz JL, Windebank AJ. (2003). Acute Glucose Deprivation Leads to Apotosis in a Cell Culture Model of Acute Painful Diabetic Neuropathy. Journal of the peripheral nervous system.
- ↑ Arumugam, Sarasa Bharathi (1988). Diabetic Peripheral Neuropathy-Insulin Neuritis-Page 5. RSSSDI.
- ↑ Llewelyn, J. G., et. al. (1986). Acute Painful Diabetic Neuropathy Precipitated by Strict Glycemic Control. Acta Neuropathica.
- ↑ American Diabetes Association-Chat Transcript. American Diabetes Association (2005).
- ↑ Starting Insulin Treatment for Type 2 Patients-Page 22. Royal College of Nursing.
- Perineurial Abnormalities in the Spontaneously Diabetic Dog Ghani M, Malik RA, Walker D, Sharma AK, Lowrie CT, Schall WD, Boulton AJ., 1999, Acta Neuropathologica
- Electrodiagnostic Analysis of Peripheral Neuropathy in Dogs with Diabetes MellitusSteiss JE, Orsher AN, Bowen JM., 1981, American Journal of Veterinary Research
Study with long-time diabetic dogs has findings similar to those of humans--that long-term diabetes tends to mean some neuropathy.
- Peripheral Neuropathy--Diabetes and Non-Diabetes Related Southpaws
- Definition of Perineurium Dorland's Medical Dictionary
- Clinical peripheral neuropathy associated with diabetes mellitus in 3 dogs Morgan MJ, Vite CH, Radhakrishnan A, Hess RS., 2008, Canadian Veterinary Journal