Neuron connectivity- how are they connected physically
Neurons are suspended, as you say, in an extracellular matrix. Brain tissues are a little bit more specific. Here I quote a few summaries from literature to answer and give your a perspective on your basic question. In bold I highlight important statements which differentiate the brain's ECM from the ECM found elsewhere in the body.


  Barros, Franco &amp; Müller, 2011: An astonishing number of extracellular matrix glycoproteins are
  expressed in dynamic patterns in the developing and adult nervous
  system. Neural stem cells, neurons, and glia express receptors that
  mediate interactions with specific extracellular matrix molecules.
  Functional studies in vitro and genetic studies in mice have provided
  evidence that the extracellular matrix affects virtually all aspects
  of nervous system development and function. Here we will summarize
  recent findings that have shed light on the specific functions of
  defined extracellular matrix molecules on such diverse processes as
  neural stem cell differentiation, neuronal migration, the formation of
  axonal tracts, and the maturation and function of synapses in the
  peripheral and central nervous system.
  
  Ruoslahti, 1996: The extracellular matrix of the adult brain
  tissue has a unique composition. The striking feature of this matrix
  is the prominence of lecticans, proteoglycans that contain a lectin
  domain and a hyaluronic acid-binding domain. Hyaluronic acid and
  tenascin family adhesive/anti-adhesive proteins are also abundant.
  Matrix proteins common in other tissues are nearly absent in adult
  brain. The brain extracellular matrix appears to have trophic effects
  on neuronal cells and affect neurite outgrowth. The unique composition
  of this matrix may be responsible for the resistance of brain tissue
  toward invasion by tumors of non-neuronal origin.
  
  Dityatev et al. 2010: The extracellular matrix (ECM) of the
  central nervous system is well recognized as a migration and diffusion
  barrier that allows for the trapping and presentation of growth
  factors to their receptors at the cell surface. Recent data highlight
  the importance of ECM molecules as synaptic and perisynaptic scaffolds
  that direct the clustering of neurotransmitter receptors in the
  postsynaptic compartment and that present barriers to reduce the
  lateral diffusion of membrane proteins away from synapses. The ECM
  also contributes to the migration and differentiation of stem cells in
  the neurogenic niche and organizes the polarized localization of ion
  channels and transporters at contacts between astrocytic processes and
  blood vessels. Thus, the ECM contributes to functional
  compartmentalization in the brain.